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THE OFFICIAL JOURNAL OF THE FIBONACCI
ASSOCIATION
VOLUME 20
NUMBER 2
CONTENTS
Ducci Processes
.....
Fook-Bun Wong
Minimum Periods Modulo n for
Bermoulli Polynomials
Wilfried Herget
A Note on Fibonacci Primitive Roots . . . . . . . . . . . . . Michael E, Mays
A Property of the Fibonacci Sequences
(Fm), m = 0 , 1 ,
L. Kuipers
Notes on Sums of Products of Generalized
Fibonacci Numbers
David L. Russell
Binetfs Formula for the Tribonacci Sequence
W. JR. Spiekerman
Pythagorean Triples . . .
. . . . . . . . . . . . . . . . . . . . . . A.F. Horadam
Composition Arrays Generated by
Fibonacci Numbers V. I?. Hoggatt, Jr., & Marjorie Bickmell-Johnson
The Congruence xn = a (mod m),
Where (n, 0 (m)) = 1,
Af. J. DeLeon
On the Enumeration of Certain Compositions
and Related Sequences of Numbers
Ch. A. Charalambides
A Generalization of the Golden Section . . . . . .
. . . . . D.H. Fowler
The Fibonacci Sequence in Successive Partitions
of a Golden Triangle . . . . . . . .
Robert Schoen
Geometry of a Generalized Simson's Formula . . . . . . A.F. Horadam
An Entropy View of Fibonacci Trees . . .
. . . . . . . . Yasuiehi Horibe
Elementary Problems and Solutions . . . . .. Edited by A. P. Hillman
Advanced Problems and Solutions . . Edited by Raymond E. Whitney
97
108
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114
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185
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1982 by
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DUCCI PROCESSES
FOOK-BUN WONG
Simon Fraser
University,
(Submitted
Burnaby, B.C.,
April
1979)
Canada
1. Introduction
During the 1930s Professor E. Ducci of Italy [1] defined a function whose
domain and range are the set of quadruples of nonnegative integers. Let
J \X -^ , X 2
9
^ 3 s *K i± )
l ~ ^2 I '
~" M
I 2 ""
3 1
s
I 3
—
1+ I »
|if
~" ^ 1 1 ) '
n
Let f (x19
x29 x3, xk) be the nth iteration of f.
Ducci showed that for any
choice of x , x 9 x , a? there exists an integer N such that
fm(x19
x29
x39
= (0, 0 9 0, 0) for all m > N.
xh)
We note the following properties of the function / of the previous paragraph:
(1) There exists a function g(xs y) whose domain is the set of pairs of
nonnegative integers and whose range is the set of nonnegative integers.
[Here g(x9 y) = \x - y \ ] .
(2) f(x19
x29 x39 xh) = {g(x19
x2)9
g(x29
x3)9
g(x39
xh)9
g(xk9
n
x±)).
(3) The four entries of f (x19
x29 x39 x^) are bounded for all n.
bound depends on the initial choice of x1, x29 x39
xh.
The
We call the successive iterations of a function satisfying these conditions a Ducci process. Condition (3) guarantees that a Ducci process is either
periodic or that after a finite number of steps (say N)
fn+1(x19
x2,
x39
xk)
= fn(x19
x29
x39
xh)
for all n > N.
If a function g generates a Ducci process of the latter type, we say that
g is Ducci stable (or simply stable).
2. Illustrations
(1) Let g(x9 y) = x+y (mod 3) 9 where x (mod 3) is the least nonnegative
integer congruent to x (mod 3). Then, an example shows that g is not stable.
Seta^ = x2 = x3 = 0 and xh = I. We may tabulate the successive values of /
as follows:
•f
f
\
f
ff
l
K
K
Kf
(0,
(0,
(0,
(1,
(1,
(1.
(2,
(0,
(1.
(0,
0,
0,
1,
0,
0,
1,
1,
1,
2,
.0,
97
0,
1,
2,
0,
1,
0,
0,
1,
1,
1,
1)
1)
1)
1)
2)
0)
1)
0)
0)
1)
98
[May
DUCCI PROCESSES
Since / 9 (0, 0, 0, 1) = ^ ( O , 0, 0, 1)= (0, 0 3 1, 1), the process is periodic with period 8 and g is not stable.
(2) Let g(x, y) = x + y (mod 8).
same initial values.
We construct a similar table for the
(0,
(0,
(0,
(1,
(A,
(2,
(4,
(4,
(0,
(0,
ff
\
K
K
K
r
K
f
f
0,
0,
1,
3,
6,
2,
0,
4,
0,
0,
o,
1,
2,
3,
4,
6,
4,
4,
0,
0,
1)
1)
1)
1)
2)
6)
0)
4)
0)
0)
We observe that for n _> 8, fn(0,
0, 0, 1) = (0, 0, 0, 0). We prove below
that g is stable, viz.., that any choice of initial values leads to a similar
result.
We now list a set of functions which can be proved to be stable. In some
cases we prove the stability of the function and in others we leave the proof
to the reader.
3. Theorem
The following functions are stable:
(1) x + y (mod 2 n ) , n - 1, 2, 3, ..,
(2) x • y (mod 2 n ) 9 n = 1, 2, 3,
(3) x* + z/* (mod 2 n ) , t = 2, 3, 4, ., ; n = l ,
2, 3, ... .
(mod 2 n ) ; , t = 2 , 3, 4, . .; n = 1, 2, 3, ... .
(4)
(x + yY
(5)
\x* - yt\
(6)
J (x - z/)* | (mod 2"), t = 1, 2, 3, . .. ; n = 1, 2, 3, . . . .
(7)
<$>(x) + §(y)*
(mod 2 n ) , t = 1, 2, 3,
..; n = 1, 2, 3, ... .
where (J) is Euler's c[)-function.
The notation ir (mod 2") means the least nonnegative integer congruent to
x modulo 2 n .
f(x19
4).
Proof of (1): We use f£ to denote the ith entry of the nth iteration of
x2, x3, xh).
The subscript i + J of x will always represent i + j (mod
We first consider the function g1(x,
y) = x + y and show that for any n:
f2(n + i)=
We compute:
(2«)[(2»
f1
= tf^ + ^
•>£
= x
i
+
_
l)Xi
+
(2» +
1}
^ +2
+*i+3)].
(A)
(B)
+ 1.
2x-i,+ i
+ 2 *(x i + 1
+
(C)
^i + 2*
(D)
f4 = 2 ^ + 4^i + 1 + 6xi+2
+ 4 ^ +3.
(E)
1982]
DUCCI PROCESSES
(A) i s c l e a r l y t r u e f o r n = 1 by ( E ) .
f-'n+2)
- #<jfn+1)>
99
Suppose (A) i s t r u e f o r n.
= ( 2 n + 1 ) [ ( 2 « + 1 - l)xi
Then by (C)
+ (2* + 1 4- l ) * , + 3
+1
+ (2n+1)(a;. + * i + 2 ) ] .
(F)
We note that for any iteration of f(x19
x29 x3, x^) we can consider (f"9
f29
f3n9 fO
to be the same row as its transpositions (/», f?9 f%9 /£) , (/», /*,
/i1, / £ ) , and (/2n, /3n, ft, ft). Therefore (F) indicates that (A) is also true
for n + 1. It follows by finite induction that (A) is true for all n.
Hence
we conclude that
/ ? ( n + 1 ) = 0 (mod 2 n ) , n = 1, 2, 3, ... .
Since fn(09
0, 0, 0) = (0, 0, 0 5 0) for all n9
g(x9 y) = x + y (mod 2n) is established.
Proof of (3): For any initial numbers x19
to arrange even and odd numbers:
the stability of the function
x29
(i)
(e9
e9 e9 e)
(iv)
(ii)
(e9
e9 e9 b)
(v)
(iii)
(e9
e9 b9 b)
(vi)
x39
xk9
there are six ways
(e9 b9 e9
b)
(e9
b9 b9
b)
(b9 b9 b9
b)
where e and b represent even and odd numbers, respectively. Since the sum of
the tth powers of two even (or two odd) numbers is even, the sum of the tth
powers of an even number and an odd number is odd,. Therefore, when we consider the function g2{x9 y) = xt+ yt, the initial arrangements (ii) and (v) yield
the following:
(e9
l
f
f
f
f
e9 e9
b)
(e9
e 9 b9
b)
(e9
b9 e9
b)
{b9
b9 b9
(e9
e9 e9
(e9
b9 b9
b)
b)
(bs
e9 e9
(b9
e9 b9 e)
b)
(b9
b9 b9
b)
e)
(e9
e9 e9
e)
and
(G)
The arrangements (i), (iii), (iv), and (vi) are included in the above operations. Thus there exists an integer m <_ 4 such that all numbers of fm(x19
xl9
x39 Xi+) are even numbers for the arrangements (i)-(vi).
Let fm(x19
x29 x39 xh) = (2im19 23m29 2um39 2vmh)9
where i9 j , u9 v9 m19
m29 m39 mh are positive integers. Without loss of generality, we may assume
that i <_ J, u, V. Then we have
( 2 ^ ) * + (2jm2)t
= 2^*777* + 2Hm2t
^ I** (m\ +
2U'i)tmt2)a
This indicates that the value of i in fm will increase by at least t times at
the next step (where t J> 2) . After a finite number of steps, we can obtain an
integer q such that fq(x19
x29 x39 x^) = (2hq19
2lq29
2rq39
28ql+)9 where h9 £,
r9 s9 q19 q29 q39 qk are positive integers and h9l9v9s
>_ n9 i.e., all numbers
of fq are the multiples of 2 n . Thus, the four numbers of fq are congruent to
zero modulo 2n. This shows that the function g(x9 y) = xt + y* (mod 2n) is
stable.
100
DUCCI PROCESSES
[May
Before we prove the last statement of the theorem, let us recall a simple
property of Euler's ^-function.
Lemma 1: For any even integer N9 (i) lf-N is a power of 2, then $(N)
(ii) if N is not a power of 2, then $(N) < (h)N.
Proof of (7):
greater than 2.
=
(k)N;
First9 we consider only the initial numbers x±9 x29 x39
Since cj)(x) is even for all x > 2. Therefore, we have
f1(xl9
x29
x39
xj
= (Nl9
N29 N3,
xh
Nh)9
where Nl9 N2, N$9 N^ are even integers and min-Oi, N2, N39 N^} _> 4. If N1 =
N2 = N3 = Ni+9 we can see below that statement (7) of the theorem is true in
this case. If all four are not equal, by Lemma 1 it is clearly seen that the
greatest integer (if two or three are equal and greater than the remaining,
each of these may be called "the greatest") of fn(x19
x29 x3, xh) must get
smaller within three steps for all n.
Hence, after a finite number of steps
(such as m)9 fm(x19
x2, x39 xh) = (N59 N6, N79 NQ)9 where N59 N69 N79 NQ are
even integers and either N5=N6=N7=N8=2t
for some integer t >_ 3, or
max{N59 N69 N79 N8} = 4. But, we also have m±n{N5, Ns, N7, NQ} _> 4 and
/ c (2*, 2*, 2*, 2*) = (2*, 2\
2*9 2*) for all c and t .
This implies that the function §(x) + $(y) is stable.
It remains only to show that the initial numbers x19 x29 x39 xk contain
some 2s or Is [since (f)(2) = <J)(1) = 1» so we only need to consider either 1 or
2]. Suppose the initial numbers contain only one number 2, say x± = 2. Thus
f1(xl9
x29
x39
- (l + cjK#2), <K#2) + <K# 3 ), <K^3) + <f>0&i>)» *(^4) + l)-
x^)
Since x29 # 3 , x^ > 2. Therefore, all four numbers of f1 (x1, x29 x39 xk) are
strictly greater than 2. Similarly, when the initial numbers contain two or
three 2s» we can prove that there exists an integer j <_ 3 such that
J
where J19 J29 J39 Jh
the proof of (7).
\X-^ ,
X29X39X^)
=
{ J ^
9
d
2 9
are. integers and m±n{J1, J29
J
3 9
t / i
J39
i
)
9
Jk}>
2.
This completes
4. Some More Ducci Processes
Let us denote the w-digit integer by
x = I0m-1cim+
I0rn'1am_1
+ . . . + 10a 2 + a1
and
S$ = (am + a w . x + . . . + a 2 + q ) * , T* = a* + a*_i + • • • + a\ + a £ ,
where t = 1, 2 , 3 , . . . .
We now address the following problems:
(1)
For what values of t is the function \s£ - Sy\
stable?
(2)
For what values of t is the function \T% -'Ty\
stable?
(3) For what values of t and n is the function T* + T$ (mod 2n) stable?
Partial answers to these questions are given below.
Obviously, the function \Sx - Sy \ is stable for t - 1.
stability for t = 2, we need the following lemma.
In order to prove
1982]
DUCCI PROCESSES
101
Lemma 2: Let Z be the set of all nonnegative integers and let # = {3s: ze Z},
L = Z\H. Then for any h9h19h2e
H and £,£ 1 9 £ 2 ££ w e nave
(i)
|/z2 - h\\ eH and | £ 2 - l\\
eH;
(ii)
Proof: For any heH, we have ft = 0 (mod 3) and ft2 E 0 (mod 3), For any
leL9
we have either £ = 1 (mod 3) or £ = 2 (mod 3). But we see that I1 = 1
(mod 3) for both cases. Therefore, we obtain
(i)
\h{ - h\\
\h\
(ii)
E 0 (mod 3) and |£2 - %\| E 0 (mod 3), i.e.9
- h\\ eH and | £ 2 - l\\
eH.
|ft2 - £ 2 | = |l|, i.e., \h2 - £ 2 |eL.
We may note that by division by three a nonnegative integer has the same
remainder as the sum of its digits. Therefore, an immediate consequence of
Lemma 2 is:
Lemma 3: Let Z be the set of all nonnegative integers and let # = {3s: s e Z } ,
L = Z\H. Then for anyft,ftx,ft2e#and l9l19l2eL
we have
(i)
(ii)
\S2K - S2J e H and | ^ - S2J e H;
\S2 - S\\
eL.
We now prove that the function \S% - Sy | is stable for t' - 2. By Lemma 3
we see that e andfccan play the same roles as shown in (G) if e represents
the initial number which belongs to the set H and b represents the initial
number which belongs to the set L. Thus we can find an integer m <_ 4 and four
integers hl9 h29 h3, hk e H such that
fm(xl9
x29
x39
xh)
= (ftl5 h29 ft3, hh)
and Sh1$ Shl9 Sh^9 S^ e H. It follows that there exist four nonnegative integers
h59 ft6, h79 ft8 such that
fm+1(xl9
x29
x39
a?lt) = (ft5,ft6,ft7,ft8)
and Shi E 0 (mod 9), i = 5, 6, 7, 8. On the other hand, if max{ft5,ft6,h79 ft8}
has four or more digits, then, after a finite number of steps (say d), we can
find four nonnegative integers ft9, ft10s h119 h12 such that
X 2 $ X 2 9 X i^
where Sh.
proof is
(9 + 9 +
that the
that
= 0 (mod 9), £ = 9, 10, 11, 12 and max{ft9, ft10, ftxl, ?z12} < 999 (the
based on the same principle as shown in Steinhaus [2]). We know that
9) = 27. Therefore, max{^ 9 , Shl0> Shll , Shl2.} < 27. This indicates
values of Sh. (i - 9, 10, 11, 12) are either 0, 9, or 18. But we see
18 2 - 0 2 = 324 and 3 + 2 + 4 - 9;
182 - 9 2 = 234 and 2 + 3 + 4 = 9; .
1 8 2 - 18 2 = 0.
[May
DUCCI PROCESSES
102
Thus, in the next step., we have
J
\Xl9
a? 3 , Xk)
XZf
=
( « 1 3 J ^iit» ™15»
^16^»
where the values of Shi (i = 13, 14, 15, 16) are either 0 or 9. It is easily
verified that fc(xl9
x29 x3i x±) = (0, 0, 0, 0) for all o >_ m + d + 5. This
shows that the function |-5| - S§ | is stable.
The function |£* - Sy\
For instance, letting
J
y I is not stable for t _> 3.
#0r, i/) = \SI -
S\\-
and
x±
= 21951, x2 = 21609, a?3 = 0, xh
= 324,
we have
(21951, 21609,
(
0,
5832,
( 5832,
0,
( 5832,
729,
f
0,
729,
5103,
5103,
324)
5103)
729)
0)
Since
3
/ (21951, 21609, 0, 324) = /1(21951, 21609, 0, 324) = (5832, 729, 5103, 0 ) ,
the process is periodic with period 2. The same result is obtained if we take
(531441, 0, 426465, 104976) as the initial entries for t = 4.
The reader is welcome to consider the stability of the function \T£ - Ty\
in problem (2). About 500 quadruples of two-digit numbers (xl9 x2, ^ 3 , x^)
have been tested for t = 2 and t - 3. In each case, the functions \T% - Ty\
and \T£ - T%\ stabilized after 80 steps.
With respect to problem (3), it is not difficult to get an example to show
that the function Tg + Tg (mod 32) is not stable. Letting
X •!
==
1U j Xn
~ Z. c-, Xo
:=
0, X h,
==
^-0,
we have
(10, 22, 6, 26)
( 9, 12, 12, 9)
(22, 10, 22, 2)
( 9, 9, 12, 12)
f
Thus, the process is periodic.
5. Ducci Processes in fc-Dimensions
By analogy with Section 1, we now consider a function / whose domain and
range are the set of ^-tuples of nonnegative integers. Suppose that there is
a function g(x9 y) whose domain is the set of pairs of nonnegative integers,
whose range is the set of nonnegative integers, and that
f{xl9
m
x29
Let f (xl9
..., xk)
= (g(xl9
x2)9
g(x2,
x3)9
..., g(xk,
x±)).
..., x^ be the mth iteration of /. Assume that entries of
) are bounded for all m (as before the bound depends on the
initial choice of entries)»
A Ducci process is a sequence of iterations of f.
We call a function g
stable if g generates a Ducci process such that for any choice of entries
x29
fm+l(xl9
x2,
..,., xk)
= fm{xl9
x29
..., xk)
for some m.
All of the Ducci processes in Sections 1-4 can be generalized to an arbitrary dimension k9 where k is any integer greater than 2. We propose to examine only two such generalizations.
1982]
DUCCI PROCESSES
103
B. Freedman [3] proved that function g(x9 y) = \x-- y\ is stable if the
number of members of the initial entries k is a power of 2.
We now show that the following functions are stable if and only if k is a
power of 2.
(I) g(x9
(II) g(x,
y) = x + y (mod 2n),
y) = x + y (mod k)
s
n = ls 2, 3, ... .
where k is an arbitrary positive integer.
Proof of (I): Let kfim be the ith entry of the mth iteration of f(xl9
x29
..., xk).
The subscript i + j of x will always represent i + j (mod &) .
Consider the function g(x9 y) = x + y. We can show by mathematical induction that for any m9
k
A'-?(")v
J-O
<H>
In fact, (H) is true for m = 19 because
k
f} = x, + x.
Suppose (H) is true for w.
=
.
Then
(oh + £[(*) + G- - i)] x *w + (IK-1
+
?>
*V •
J'-O
Therefore, (H) is true for all m.
In particular, if fe is a power of 2 (fc = 2 r ) , then from (H) we have
2
/->r\
/0r>
21--l/0rx
2r-l ,
= 2tf; +
•
3 *
T
Adopting Freedman s techniques we see that ( . ) is always even for J = 1,
.., 2r - 1. Hence
*/* =• 0 (mod 2), i = I, 2, ....,fc,fc= 2r
104
DUCCI PROCESSES
[May
and
*/?* =-0 (mod 4), £ = 1, 2, ..., k, k = 2 2
In general
fe^tk
:
/.
= 0 (mod 2*), £ = 1, 2 9
j rC , K, "~ i.
.
Thus, we conclude that for any n we have
*/,"* = 0 (mod 2"), £ = 1, 2, ...,fc,k = 2*\
This means the function ^(x, zy) = x + zy (mod 2n) is stable if k is a power of
2.
Proof of (II): The function g(x9 y) = x + y (mod k) is stable if and only
if k = 2 P for any p. That this condition is sufficient follows from the previous proof. We now show that it is necessary.
We prove first that g(x, y) is not stable if k is an odd prime p. Let the
initial entries be xl9 x29 ..., xp, and
_f0
if 0 < £ < p
if % - p.
Then from (H) we have
if 0 < £ <
VryV
y,
if £ = p.
We know that f
Hence,
. =
.
= 0 (mod p) for 0 < £ < p when p is an odd prime.
PfP
h
=
f 0 if 0 < £ <
[2
if £ = p
and
P/P*
_ f 0 if 0 < i < p
2* if £ = p
,
, .
where t is a positive integer. Thus, by Fermat's theorem 2P~1
obtain
PfP(p-i)
_ J °
i f 0 < £ <
^
[ 1 if £ = p.
= 1 (mod p ) , we
Therefore, #(#, z/) is periodic.
Now let k = ps9 where p is an odd prime and s is any integer greater than
one. Let
if £ is a multiple of p
otherwise.
*< - {l
*For example, let k = 6; then k = ps
as (0, 0, 2, 0, 0, 2). Thus we have
6J?1
6^*3
f>/6
(0,
(0,
(2,
(0,
(0,
(4,
(0,
0,
2,
4,
0,
4,
2,
0,
2,
1,
2,
4,
4,
4,
2,
3X2.
0,
0,
2,
0,
0,
4,
0,
0,
2,
4,
0,
4,
2,
0,
2)
2)
2)
4)
4)
4)
2)
We set the initial entries
1982]
DUCCI PROCESSES
105
Then
knP
J i
and
k_ppt
•* i
•{
2 s
0
where t is a positive integer.
fc«p(p-D
•^
Thus, function g(x9
if i is a multiple of p
otherwise
2s
0
if i is a multiple of p
otherwise
,
(m
°
d
, -x
k)
'
Hence
J s if £ is a multiple of p
1 0 otherwise.
y) is periodic and the proof is complete.
We leave it to the reader to examine generalizations of the Ducci processes presented in Sections 1-4.
References
1.
2.
3.
R. Honsberger. Ingenuity
in Mathematics.
New York: Yale University, 1970,
pp. 80-83.
H. Steinhaus. One Hundred Problems in Elementary Mathematics.
New York:
Basic Books, 1964, pp. 56-58.
B. Freedman. "The Four Number Game." Scripta
Mathematica
14 (1948):3547.
*****
MINIMUM PERIODS MODULO n FOR BERNOULLI POLYNOMIALS
Technlsche
J1.
WILFRIED HERGET
Universitat,
Clausthal,
Fed. Rep.
(Submitted
September
1980)
Germany
It is known that the sequence of the Bernoulli numbers bm9
defined by
b0 = 1,
is periodic after being reduced modulo n (where n is any positive integer),
cf. [3]. [In this note, we use the symbols bm for the Bernoulli numbers and
Bm(x) for the Bernoulli polynomials.] In [3] we proved
Theorem 1: Let p e /P, IP being the set of primes, p il 3, and e9k9m
k9m >_ e + 1, we have:
bk n-integral and k = m mod pe(p
e IN.
For
- 1) =>bm n-integral and b\ = bm mod pe.
In this n o t e , w e shall give some analogous results about the sequence of
the Bernoulli polynomials Bm(x) reduced modulo n (Theorem 6) and the polynomial functions over Zn generated by the Bernoulli polynomials (Theorem 4 ) .
H e r e , Zn is the ring of integers modulo n , where n e IN, n >_ 2, and the B e r noulli polynomials in Q[x] are defined by
i =oVv I
Similar questions about Euler numbers and polynomials were asked by Professor
L. Carlitz and Jack Levine in [2].
j2. In [4] we discussed in which cases it is possible to define (in a natural
way) analogs of Bernoulli polynomials in Zn.
In this section, we shall prove
the periodicity of the sequence of the polynomial functions Bm over 2n generated by the Bernoulli polynomials. Each polynomial F(x) e Q[x] generates a
polynomial function F : 2 ->- Q by
(1)
x *
N o w , considering (1) in Zn9
(a)
(b)
F{x).
w e get a function F : Zn •> Zn if and only if
all values of F are interpretable mod n, and
the relation (1) preserves congruence properties.
For this, it is useful to introduce the following notations ([4], p. 28).
Definition 1:
A function F : 1 -> Q is said to be acceptable mod n, iff
(a) V x9 F(x) is n-integral,
(b) x E y mod n => F(x) = F(y)
mod n.
106
[May 1982]
MINIMUM PERIODS MODULO n FOR BERNOULLI POLYNOMIALS
A polynomial F(x) e Q[x]
its polynomial function.
Definition 2:
107
is said to be acceptable mod n if this is true for
Two functions F9G : Z -> Q are said to be equivalent mod n iff
(a) F9G are acceptable mod n,
(b) V x9 F(x) = G(x) mod n.
Two polynomials F(x)9G(x)
e Q[x] are said te be equivalent mod n if this is
true for their polynomial functions. We write
F ~ G mod n
and
~~ G(x) mod n,
F(x)
respectively.
From [4], p. 29, we have the following
Theorem 2: Bm(x)
is acceptable mod n
o bm is n-integral and rnSm_1(n)
,.
where
x-1
£
$«<*>
= 0 mod n,
km for 7w £ {0, 1, 2, . . . } , # e N
fc = 0
for w = -1 or x = 0.
0
A more explicit characterization of Bm(x)
p. 31)
acceptable mod n gives (cf. [4],
—
Theorem 3:
(a)
For m > 1 and 2 1 n we h a v e :
Bm(x)
a c c e p t a b l e mod n
<* V p £ /P : ( p | n =* p - 117?7 and (p - 1 /f m - 1 or p 177?) ) .
(b)
For k e IN we have:
Bm(x)
acceptable mod 2k o m = 0.
Now, we may state our first new assertion.
discuss the case n = pe9 p a prime, p >. 3.)
Theorem 4: Let p e IP, p _> 3, and e9k9m
(a)
(By Theorem 3, it suffices to
e E.
For k9m _> e + 1, we have:
Bk
acceptable mod p and k = m mod pe (p - 1)
=» 5 W acceptable mod p and J3fc ~ Bm mod p e .
(b) p&(p - 1) is the smallest period length of the sequence of the Bernoulli polynomials in the sense of (a).
For the proof of this theorem, we need the following
Lemma: Let p e p, p _> 3, e e IN.
Then
a.ACpO + vep') = XV(P')
where both F(pe) = 5 and X(pe)
= pe~1(p
mod
pe
for
all
^
- 1) are minimal for this property.
For the proof of this lemma, see [5], Theorem 1.
108
MINIMUM PERIODS MODULOn FOR BERNOULLI POLYNOMIALS
[May
Proof of Theorem 4(a): Let k9m > e, k = m mod pe(p - 1) and Bk be acceptable mod p, so that b\ is p-integral and kSk_1(p)
= 0 mod p. By Theorem 1,
bm is p-integral with bk =• bm mod pe.
Furthermore, from k = m mod pe(p - 1),
and k9m > e9 we have k° ik~x ^ m • i™'1 mod pe for all i, by the lemma above.
Then
kBk_1(x)
for all x.
»• kj^i1"'1
= m ^ i ™ " 1 = /7^m-1(a:) mod p e
Now we use (5) from [4]:
Bm(x) = mSm^1(x)
+ iOT.
= B w (#) mod p e for all x% i.e.,
Thus £ m is p-integral, too, and Bk(x)
B k - Bm mod p*.
Proof Of Theorem 4(b): Let Bk and Bm be acceptable mod p, let k9m _> e+ 1,
and let B k ~~ BOT mod p e . Then Bk(x) = B m (#) mod p e for all #. We shall show
that k = m mod pe(p - 1) if pe)( mm Obviously this would prove the assertion.
First, we get bk - Bfc(O) = B m (0) = bm mod p e , hence kSk_1(x)
= mSm_1(x)
mod p e
for all x; and moreover,
= mx"1'1
fcc^"1
(2)
mod p e for all x9
since
kxk'1
= kSk_1(x
+ 1) - feS^Cs).
e
Putting x = 1 in (2) shows fc = m mod p . Let d = g.c.d.(fc, p e ) . We know that
g.c.d.(m9
pe) = d9 and d ^ pi with 0 <. £ < e, since p e /f ?C. Thus (2) implies
x k _ 1 = x""1 mod p*'1
for all x.
But this is possible only i f f c - l = m - l mod (p - 1); i.e., Jc = m mod (p- 1).
Together with k = m mod p e , we have k = m mod plS(p - 1), and the theorem is
proved.
Remark 1: The minimum period length of the Bernoulli polynomial functions mod
n is the same as that of the Bernoulli numbers mod n.
Remark 2: By a very similar argument one may prove that when Bm is acceptable
mod p, m E 0 mod pe o Bm ~ 0 mod pe.
For this, notice that m = 0 mod p e ime
plies fcm = 0 mod p ([l],p. 78, Theorem 5).
Remark 3: Let v(pe)
denote the preperiod length of Bm mod pe.
Then Theorem 4
implies v(pe)
<_ e + 1. Using Remark 2 one may slightly improve this inequality for special cases with e >_ p. For instance, u(33) = 3.
_3. In this section we shall discuss the periodicity of Bernoulli polynomials
reduced modulo n.
Definition 3: A polynomial F(x) = a0 + axx + ••• + arxr e Q[x] is said to be
n-integral if and only if the coefficients a 0 , a19 .. ., ar are all n-integral.
From [4], p. 32, we have, for the Bernoulli polynomials,
Theorem 5:
Let p e P9 e e M9 and m e ffl U {0} with p-adic representation
s
m = E
m
fc = 0
kPk-
1982]
MINIMUM PERIODS MODULO n FOR BERNOULLI POLYNOMIALS
109
Then
s
Bm(x)
e Q[x] is pe-integral if and only if J : ^
< p - 1,
fc = o
Remark 4: Each n-integral polynomial is acceptable mod n, but there are polynomials acceptable mod n that are not n-integral (cf. [4], pp. 32-33). If we
reduce the coefficients of any n-integral Bm(x)9
we still get a polynomial of
degree m, since the coefficient of xm is 1. Consequently, no periodicity appears. But by the lemma above we have
e
xP -Hp-l)
+e
xe
E
mod
ve
al]_
for
x>
Hence, any p-integral polynomial F(x) is equivalent to a reduced polynomial
with degree < p6"1^
- 1) + e having coefficients in {0^ 1, ..., pe - l}. We
shall denote such a polynomial F(x)9 reduced mod n, by F(x).
Remark 5: If F (x)
are reduced polynomials of F(x)
and F2(x)
F1(x)
~ F2(x)
- F(x)
mod n, then
mod n.
We conjecture that the sequence of the Bernoulli polynomials, reduced mod
n, is periodic in a strong sense too, with a proof here only for n - p9 p e P .
Theorem 6: Let p e P, k9m _> 2, and suppose Bk(x),
k = m mod p{p - 1), then
Bk(x)
= Bm(x)
in
Bm(x)
are p-integral.
If
lv\x\.
Proof: Bk(x)9
Bm(x) p-integral implies Bk{x)
mark 4). By Theorem 4 we get
9
Bm(x)
acceptable mod p (Re-
mod p, hence
Bk(x)
~~ Bm(x)
Bk(x)
~~ Bm(x)
Bk(x)
- Bm(x)
mod p, i.e.,
= 0 mod p for all x.
The degree of this difference polynomial is <X(p) + V(p) = p - l + l = p , but
it has p zeros in Z p , hence it must be the zero polynomial, and we have
Bk(x)
= Bm(x)
in
Zp[x].
Remark 6: The question, whether Theorem 6 holds for arbitrary modulus n, remains open. The proof above fails in ln when n $. F9 since Bk (x) ~~ Bm(x) mod n
does not imply Bk(x) = Bm(x) in ^„ |>]. For example, let e > 1 and
p-i
F(X)
= p6-1 n (^ - ^)5
i=o
pe-
i
£(*) - II (o? - i ) .
t=o
Then F(x) - G(x) (-C) mod p e , but F(tf) + G(x) in %pe[x].
Or, if n = p ^ ,
where p19p2
£ ^ and p 1 ^ p 2 , then one may consider the polynomials
Px-i
P2
n (# - ^)
t«0
for a counterexample.
and
P2-1
Pi n & - ^)
t - 0
110
MINIMUM PERIODS MODULO n FOR BERNOULLI POLYNOMIALS
Remark 7: The assumption in Theorem 6 that both B^{x) andBm(x)
cannot be weakened, since B^(x) p-integral and k E m mod p(pply Bm p-integral. For example
B2(x)
is 5-integral, while B22(x)
5 • 4.
= x2
[May 1982]
are p-integral
1) does not im-
- x + -7-
is not so by Theorem 2, even though 22 = 2 mod
References
1.
L. Carlitz. "Bernoulli Numbers." The Fibonacci
Quarterly
6, no. 3 (1968):
71-85.
2. L. Carlitz & J. ^evine. "Some Problems Concerning Kummerfs Congruences for
the Euler Numbers and Polynomials." Trans. Amer. Math. Soc. 96 (1960):2337.
3. W. Herget. "Minimum Periods Modulo n for Bernoulli Numbers." The
Fibonacci Quarterly
16, no. 6 (1978):544-548.
4. W. Herget. "Bernoulli-Polynome in den Restklassenringen %n."
Glasnik Matematicki
14, no 34 (1979):27-33.
5. D. Singmaster. "A Maximal Generalization of Fermat's Theorem."
Mathematics
Magazine 39 (1966):103-107.
*****
A NOTE ON FIBONACCI PRIMITIVE ROOTS
West Virginia
MICHAEL E. MAYS
University,
Morgantown,
(Submitted
November 1980)
WV 26506
A prime p possesses a Fibonacci primitive root (FPR) g if g is a primitive
root (mod p) satisfying
g2 = g + 1 (mod p ) .
This definition was given in [2], and properties of FPRs were worked out
in [2] and [3]. A good discussion of FPRs is contained in [4].
In [2] an asymptotic density for the set of primes having a FPR in the set
of all primes was conjectured, and that this density is correct subject to a
generalized Riemann Hypothesis was shown in[l], but it is still an open question as to whether or not infinitely many primes possess FPRs. The purpose of
this note is to provide a sufficient condition that a prime should possess a
FPR.
Theorem:
If p = 60fc - 1 and q = 30k - 1 are both prime, then p has a FPR.
Proof: p E 3 (mod 4) implies that at most one of {a, -a} is a primitive
root of p for any a such that 2 <_ a <_ (p - l)/2=q;
q prime implies that there
are q- 1 primitive roots of p in all, so exactly one of {a9 -a} is a primitive
root of p.
p E -1 (mod 10) implies that two solutions to the congruence
x2
- x - 1 E 0 (mod p)
exist. These solutions may be written as g and 1 - g.
Shanks points out that since g2 - g - 1 E 0 (mod p ) ,
g(g - 1) E 1 (mod p ) ,
so that g is a primitive root iff g- 1 is a primitive root. # - 1 is a primitive root iff ~{g- 1) = 1 - g is not a primitive root. Thus exactly one of the
solutions to the congruence is a FPR of p.
Conditions similar to that in this theorem occur frequently in theorems in
the literature about existence or ordering of primitive roots. Theorems 38-40
in [4] are well-known instances of this. In [3] it is observed that primes p
satisfying sufficient conditions to have two sets of three consecutive primitive roots (a FPR g9 g- 1, and g- 2, and -2, -3, and -4) must be of the form
120k - 1, with 60/c - 1 also prime. Using the theorem above, it is not necessary to presuppose that p has a FPR.
References
1.
2.
3.
4.
Ho W. Lenstra, Jr. "On Artinfs Conjecture and Euclid's Algorithm in Global Fields." Inventiones
Mathematicae
42 (1977):201-224.
Do Shanks. "Fibonacci Primitive Roots." The Fibonacci
Quarterly
10, no.
2 (1972):163-168, 181.
Do Shanks & L. Taylor. "An Observation on Fibonacci Primitive Roots." The
Fibonacci
Quarterly
11, no. 2 (1973).159-160.
D. Shanks. Solved and Unsolved Problems in Number Theory,
2nd ed. New
York: Chelsea, 1978.
*****
111
A PROPERTY OF THE FIBONACCI SEQUENCE (Fm),
m=
0, 1,
L. KUIPERS
Rue d'Orzival
4, 3960 SIERRE,
Switzerland
(Submitted
November 1980)
It Is well known that the sequence of the (natural) logarithms reduced mod
1 of the terms Fm of the Fibonacci sequence are dense in the unit interval.
See [1], [2]. This is also the case when the logarithms are taken with respect to a base b, where b is a positive integer _> 2. In order to see this,
we start from the fact that:
log Fn+1
Now log
- log Fn •> log
1 + /5~
as n
1 + /5,
-z /log b is an irrational number, for if we suppose that
log
£
/lo§
b = p
/s>
where v and s are natural numbers, then we would have
= ((1 + / 5 ) / 2 ) s ,
b'
obviously a contradiction., Hence, log^i^+i - log^Fn tends to an irrational
number as n -> °°. This implies that the fractional parts of the sequence
( l o g ^ ) , m = 1, 2, ...
is dense in the unit interval.
We assume that the Fibonacci numbers Fm9 m _> 1, are written in base b,
that is,
Fm = a0bn +'a1bn~1'
+ ••• + an9
where a 0 _> 1, 0 £ a3- <_ b - 1, j = 0, 1, — , n, w? = 1, 2, . . . , or to any m a
set of digits {a 0 , a l s ..., an] is associated.
Now, given an arbitrary sequence of digits {a 0 , a 1 5 ..., a r } , one may ask
whether there exists an Fm which possesses this set as initial
digits.
The
question can be answered in the affirmative.
We associate to the sequence {aQ, al9 ..., ar} the value
a,
CLy,
v + —t> +' *' • ' •» r»
which is a point on the interval [1, b).
the interval
a1
ar
T = T(
[
The function log,a:, mapping
onto the interval
This value is the left endpoint of
ax
ar
ar + 1\
[1, £>) onto [0, 1), maps this interval
a subinterval of [0, 1).
112
T(r)
[May 1982]
A PROPERTY OF THE FIBONACCI SEQUENCE {Fn)9.m
= 0 9 1, ...
113
Since the fractional parts of the logarithms to base b of the numbers Fm
are dense in the unit interval, there is an m such that logbFm (mod 1) e T*.
It follows that there exists a positive integer n >_ r such that
(
a1
a2
av
a7
Hence, there exists an integer k _> n such that
a
loghFm - fe + l o g j a 0
i
|
+ E
f l 0 + l r +
-1
a 0J* + a^*
i
ar
-+ . . . + _ + . . .
. . . +pr+...
an\
+_j,
+-
+ . . . + a 2 .fc f c ".+ .... + a * " n .
References
R. L. Duncan. "An Application of Uniform Distributions to the Fibonacci
Numbers." The Fibonacci
Quarterly
5, no. 2 (1967):137-140.
L. Kuipers. "Remark on a paper by R. L. Duncan Concerning the Uniform Distribution Mod 1 of the Sequence of the Logarithms of the Fibonacci Numbers." The Fibonacci
Quarterly
7, no. 5 (1969):465, 466, 473.
*****
NOTES ON SUMS O F PRODUCTS OF GENERALIZED FIBONACCI NUMBERS
Bell Laboratories,
DAVID L. RUSSELL
Crawfords Corner Road, Holmdel, NJ 07733
(Submitted November 1978)
The infinite sequence {sn} is a sequence of generalised
Fibonacci
numbers
(also called a generalized
Fibonacci
sequence or simply Fibonacci sequence) if
s - sn_i + sn-i f° r a H n* A particular Fibonacci sequence is completely specified by any two consecutive terms. In this paper we let {'fn}9 {^n}, {hn}>
and {kn} represent generalized Fibonacci sequences, and we let {Fn} represent
the sequence of Fibonacci
numbers defined by Fn = Fn_1 + Fn_2, F0 = 0 , F1 - 1.
Theorem: If {/n}, ign^» a n d ihn} are Fibonacci sequences, then the following
summations hold for y _> x.
(1)
2-f
x
E
fi + r "
lfn+r+2ln=x.i
<. i i y
-.n = y
J~i + r$i + s ~ L j n + p ^
n +s+1
+
Tn+r+i9n+s^n
= x-1
x<_i<_y
{-*'
2-J Ji + r^-L + s'^i + t
x <_ i <_ y
=
>-Jn + r + ign
+ s + 1 "-n + t
fn.+ r&n+.s+l
•+
Jn
n+t+1
Jn + r9n + s^n+t^n
+ r+i9n
+
s "-n+t + l
~ Jn+r+l@n+s
+ l^n+t
+l
= x - 1"
Proof: The proofs are by induction on y. As base cases we take y=x- 1;
the summations are empty, and the right-hand sides vanish identically. The
induction steps are as follows:
2-J
J~i + r
x<_i<_y + l
~~ Z-* Ji + r
x±i±y
=
rf
-in-y
Jrc+ r+2Jn
= x- 1
L
Jz/ + r + l
=
Ljy+ifi + 2 + fy + r+lJ
~
Lj"^ + 2 , + 3 J
2-f
Ji + r$i+s
x <i <_y + l
~ LJ ( a : - 1 ) +P + 2-I
~ L j ( # _ ! ) + P + 2-J
]n^
= [f
2
Jy+r + 1
+1
.
2 - r fi + r^i+s
#_< i<_ z/ •
=
fy+r+l^y
r
+s+ l
-i"
Ljn + r ^ n + s + l "*" Jn + r+lGn
+ s-l
n
=
#
= x- 1
J y + r + l9y + s +1
(continued)
114
[May 1982]
NOTES ON SUMS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS
1-^ +^
L
+8+1
+
^ ( x - l)+r&
(x-l) + s+l
~
\-Jy + [email protected]+ 8 + 1
=
[f
a
L
^ n+r^n+s
\-J(x-
+
fy+r+l&y+8+l)
+
Jy.+ r+
l)+r$(x-i)
(fy + r+l3y + 8
?(x-l)
+
+ r + l$(x-l)
+S-*
J(x- 1) +r + l&(x-l)
+s -I
115
fy+-r+l^y+s+l^
lGy+s+2^
+s+l
+ f
nn=y+l
+1
*n+ v+ l^n + s -• n =x - 1*
3. We note that, as in the proofs of (.1) and (2), the bottom limit of the
right-hand side is unchanged, and we need only prove the following identityrelated to the top limit y:
\
)
^-Jy + r + l^y+s
+
+ l^y + t
Jy+r+i^y+s+l
~ Jy + r+2^y
Jy + v9y + s + Vly+1
~ Jy + r$y + sny + t*
y+t+l
Ljy + r+29y+s+2&y+t
+
Jy+r+l^y+s^y+t+l
+l
+s + 2^y + t+2
+
Jy+r+2&y
+1
fy+r+i@y+s+i
+s + Y1 y+t + 2
+
Jy + r+l&y
y+t+i
+ s+2^y
+ t+2
~ Jy + r+ \$ y + s + l"' y +t+l -» *
As a shorthand notation we let (aba) stand for fy + r+a9y+ s+b^ly + t + c> where
a, b9 c are 0, 1, or 2. Then the identity (*) can be written as follows:
(**)
(110) + (101) + (Oil) - (111) - (000) + 2(111)
= (221) + (212) + (122) - (222) - (111).
The following identities are easily verified, and the validity of (**),
and therefore of (3), follows immediately:
(221) = (001) + (011) + (101) + (111),
(212) = (010) + (110) + (011) + (111),
(122) = (100) + (110) + (101) + (111),
(222) = (000) + (001) + (010) + (011) + (100) + (101) + (110) + (111). ••
Identity (1) is well known, although it is usually stated in terms of
Fibonacci or Lucas numbers with limits of summations 1 to n or 0 to n.
Identity (2) is a generalization of the identities of Berzsenyi [ 1] . This
is easily shown using the following identity, which is easily verified, where
{fn} and {gn} are generalized Fibonacci sequences:
<4>
fn+k9m-k
- fn9m
= ( - 1 ) " (fk Gm _ n _ k
~
fo9m-n)'
We have the following:
^
Z - / Ji9i
+ 2m + b ~ U {
+
\9i + 2m + b
*i $i> + 2m + b+l -J ^ 0 _±
116
NOTES ON SUMS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS
^^m + b+i^m
+i+ l
^
'
^m + b-l^m
? Q& 2m + b-l'
+ fm+b + i9m + i + l ~ v 1 ) \fm + bGm + l ~ JoG2m+b+l)
r
-I i,_x
7*
L2jm+fc+i^m+^
+1
[May
-. i = n
+ (-1) \fm+h.1gm
- fm+bgm+1
+ Jo^m+P-U^-i•
Applying the limits, we obtain the following expression:
2^
Ji &i+ 2m+ b ~" J~n + m+byn + m + l ~ Jm + b -l@m
0< i <_n
(I + ( - i ) n \ r I
2
,
+ -P
>
>
^m + b^m + 1 i~ J 0#2m+ M '
)^m + b-l^m
which is exactly the expression obtained in [1] for even or odd n and b - 0 or
1.
The advantages of identity (2), besides its attractive symmetry, are that
(a) only a single case is needed instead of four separate cases and (b) it is
applicable to general limits, not just the sum from 0 t o n . In addition, (2)
applies to the sum of the product of terms from different
generalized Fibonacci sequences, as opposed to the original form in [1].
It should be noted that (4) can also be applied directly to (2), leading
to the summation
n+r
nn^y
_
n=x - 1
The summation of (2) has also been considered by Pond [1], whose result is
valid for the sum of the products of terms from identical
generalized Fibonacci
sequences:
-\n = y
l~i
L*
x<i±y
fifi+a
=
lT^fl-3/n/n+l
L
+
F
sfn+2^\
Jn-*-l
Recall that {Fn} is the sequence of Fibonacci numbers. This result is easily
+
derived from (2) by use of the identity fn + r = Fr-ifn
.^r/n + i*
Identity (3) has been considered by Pond [2], again in a simpler context;
he requires all three generalized Fibonacci sequences to be identical and derives the following expression:
^
2 - / f<C fi+rfi
x<i<y
+s ~ I
s
T
~
s-1 r - l '
'
'
fn-i
*>s + r+ n + lfn Jn+ l l _
n
" * '
1
'
where D(-l)n
= fn_1fn+1
- fl = (-l)n(f.1f1
~ fl).
It is not hard to show that
this summation is a consequence of (3).
The advantages of identity (3) again lie in its pleasing symmetry, its applicability to general limits, and the fact that the summation is valid for
products of different
Fibonacci sequences.
A general methodology for finding summations of the form of identities (1),
(2), or (3) has been discussed elsewhere [3]. This methodology expresses the
sum of the products of terms from several sequences, each defined by a linear
recurrence, as a standard
sum, defined below.
1982]
NOTES ON SUMS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS
117
If the sum to be found is
/ ^
•'ls n + rxJ 2, n + rz
x ^_n<_y
*'*
Jm!)n + rm
where the m sequences {/iK . .., {fm} are defined by linear recurrence relations (i.e., not necessarily Fibonacci sequences), the standard sum is a linear
combination
n = y
-I n = x - 1
with the following important properties:
1.
each term of the standard sum is the product of m terms, one from
each of the original sequences in the product to be summed;
2.
the w-tuples (ix, ..., im)
3.
the coefficients a^lt ..., im are constant and only a bounded number
of the coefficients are nonzero.
have constant integer components;
Of interest is the result that a standard sum for the sum of the products
of terms from recurrence sequences does not always exist. In particular, the
sum
2-r Jn + rGn + s^n + t^n-^u
(with {fn}s
ign}9 {hn}, and {kn} Fibonacci sequences) cannot be expressed as a
standard sum. For details, the interested reader is referred to [3].
References
1.
2.
3.
G. Berzsenyi. "Sums of Products of Generalized Fibonacci Numbers." The
Fibonacci
Quarterly
13, no. 4 (1975):343-344.
J. C. Pond. "Generalized Fibonacci Summations." The Fibonacci
Quarterly
6, no. 2 (1968):97-108.
D. L. Russell. "Sum of Products of Terms from Linear Recurrence Sequences."
Discrete
Mathematics
28 (1979):65-79.
This work was performed at the University
of Southern
California,
BINET'S FORMULA FOR THE TRIBONACCI SEQUENCE
W. R. SPICKERMAN
East
Carolina
University,
Greenville,
(Submitted
April
1980)
1.
NC 27834
Introduction
The terms of a recursive sequence are usually defined by a recurrence procedure; that is, any term is the sum of preceding terms. Such a definition
might not be entirely satisfactory, because the computation of any term could
require the computation of all of its predecessors. An alternative definition
gives any term of a recursive sequence as a function of the index of the term.
For the simplest nontrivial recursive sequence, the Fibonacci sequence, Binet's
formula [1]
_
un = (l//5)(a"+1 - B"+1)
defines any Fibonacci number as a function of its index and the constants
a = |(1 + /I)
and
3=-|(l-/5).
In this paper, an analog of Binet*s formula for the Tribonacci sequence
1, 1, 2 , 4 , 7, ..., un+1
= un + un_x
+ u n _ 2 , ...
(see [2]), is derived. Binetfs formula defines any term of the Tribonacci
sequence as a function of the index of the term and three constants, p, a,
and T.
2. .6inet's Formula for the Tribonacci Sequence
Binetfs formula is derived by determining the generating function for the
difference equation
U
n +1 =
U
n
+
U
+
n-1
U
n
n-2
t
2
'
00
Let f(x) = u 0 + u-^x'+ u2x2
function; then
+ ••• + unxn
4- •-•• = £ uix%
•
£-o
(1 - x - x2 - x3)f(x)
be the generating
= 1,
so
f(
nX)
V
1
=
i
^ . ^2 _ ^3
_
1
(1 - PE)(1 " OX) (1 - TX)
118
=
1
P(X)'
[May 1982]
BINET'S FORMULA FOR THE TRIBONACCI SEQUENCE
The roots of p(x)
of
119
= 0 are 1/p, 1/cr, and 1/T, where p, a, and x are the roots
p(:~) = ^ 3 " ^ 2 - * - 1 = 0,
Applying CardanTs formulas to pi — 1 = 0 yields
P = j ( ^ 1 9 + 3/33" + s/l9 - 3/33" + l),
a = |(j> - ^ 1 9 + 3/33 - ^ 1 9 - 3/33" + V3i .^19 +"3/33" - > 1 9 - 3/33"])5
and
T = a, the complex conjugate of a.
Approximate numerical values for p, a, and "a are:
p = 1.8393, a = -0.4196 + 0.6063i, a = -0.4196 - 0.6063^.
Since the roots of p(x)
= 0 are distinct, by partial fractions
= ,.
= i1 _- ^pa;+ i1_-^era+
^ \ /1 -—osc)(l
^ w i - —(1 - px)(l
ax)\
fW
1 - ox'
Here
A
-
'~
-
(••
1
('•
and
1
)
P/
-.
a2
(a -- P)(a •- a ) !
- f )(• •- 1 )
1
P
~
a)(p
•- a)'
(P -
a/
a2
(>-§)('-§)
.©-p)ff-«'
Consequently,
,2
/Or) =
^
oo
9
— £p*^ +
. ... (p - a)(p - a)^ = o
i =0
\(p - a)(p - a)
— 2ai^+-z:
(a - p)(a - a)^ = °, ,
(a - p) (a - a)
^~z
(a - p)(a - a)
Thus, BinetTs formula for the Tribonacci sequence is
u
-
pn+2
^
tL
+
(p - a)(p - a)
an+2
(a - p)(a - a)
_,_
+
Eav
( a - p ) ( a - a f =0
an+2
(a - p)(a - a)
120
BINET'S FORMULA FOR THE TRIBONACCI SEQUENCE
[May 1982]
Multiplying the numerators and denominators of the last two terms by (p - a)
and (p - a), respectively, yields
„.-^eHi-+
(P-a)a"+2
|p - a|2 -2£J(a)|p - a\2
+
(p-a)a"+2,
2iJ(a)|p - a|2
Using the relations O = p(cos 0 + i sin 6 ) ,
- r n(. cos
0n
n
Q + i sin n 0), e = tan*1 (l(a)/i?(a))
and combining terms:
u„ =
I
p2
-
12
, r(r
pMn + —
I p - aJ
- 2p cos 6) „n
— P cos n n0
12
I
I p - aI
, r2
cos 0 - pr(l - 2 sin2 0)
sm 0 p - a
„
3?" s m n
2
Denoting the coefficients of p n , r n cos n 09 and vn
respectively, yields
sin n 0 by a,
3, and y,
un = ap n + rn((3 cos n 0 + y sin n 0).
Approximate values for the constants are:
p = 1.8393,
0=124.69° ,
r = 0.7374,
a = 0.6184,
3 = 0.3816,
y = 0.0374.
3. An Application
ap n
Since \r\
when
= .7374 < 1, the nth Tribonacci number is the integer nearest
|pn(3 cos n 0 + y sin n 0) | < -y.
Using calculus, the value of |$ cos n 0 + y sin n 0| is at a maximum when
n0 = 5.60° + &TT, for fe an integer.
Consequently,
|pn(3 cos n 0 + y sin n 0| < — for n > 1,
Since [a + .5] = 1 (where [ ] is the greatest integer function), a short form
of the formula that is suitable for calculating the terms of the Tribonacci
sequence is
un = [apn + .5] for n >_ 0.
References
1.
2.
Vorob*ev, N. The Fibonacci
Numbers.
Boston: Heath, 1963, pp. 12-15.
Feinberg, Mark. "Fibonacci-Tribonacci." The Fibonacci
Quarterly
1, no. 1
(1963):71-74.
PYTHAGOREAN TRIPLES
A. F. HORADAM
University
of New England, Armidale,
(Submitted
June 1980)
Australia
Define the sequence {wn} byCD
w0 = a, w1 - b, wn+2
= wn + 1 +wn
(n integer >_ 0; a, b real and both not zero).
Then [2],
(2)
<» n *>n + 3> 2
+
<2Wn+A+2>2
Freitag [1] asks us to find a cn9
(3)
<^„+3'
W
= <*>» + ! +
»+2>2-
if it exists, for which
2F
n+lFn+2>
°n)
is a Pythagorean triple, where Fn is the nth Fibonacci number.
show that on = F2n+3.
Earlier, Wulczyn [3] had shown that
(4)
(LnLn+39
2Ln+1Ln+2,
It is easy to
5F2n+3)
is a Pythagorean triple, where L n is the nth Lucas number.
Clearly, (3) and (4) are special cases of (2) in which a = 0, b = 1, and
a = 2, 2? = 1, respectively. One would like to know whether (3) and (4) provide the only solutions of (2) in which the third element of the triple is a
single
term. Our feeling is that they do.
Now
(5)
W
SO
n
+
= aFn-l
1
bFn,
z
' {a + b )F2n+3
(6)
+ {lab -
{b2 + 2ab)F2n+3
b2?2n+s
-
0
if a = 2b
(7)
a
+ (a2 - 2ab)F2n+2
• if a
tt2F2n+3
a2)Fln+2
if b = 0
F
2n + 1
5a2F2n+1
if b = -2a
II
whence
' bFn
(8)
wn
—
or
bLn
or
-aLn-i
•<
oFn_x
results which may be verif ied in (5).
121
1 I
by
122
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
[May
Therefore, only the Fibonacci and Lucas sequences, and (real) multiples
of them, satisfy our requirement that the right-hand side of (2) reduce to a
single
term.
References
1.
2.
3.
H. Freitag. Problem B-426. The Fibonacci
Quarterly
18, no. 2 (1980):186.
A. F. Horadam. "Special Properties of the Sequence wn(a, b; p, q)." The
Fibonacci
Quarterly
5, no. 5 (1967):424-434.
G. Wulczyn. Problem B-402. The Fibonacci
Quarterly
18, no. 2 (1980):188.
*****
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
V. E. HOGGATT, JR.
San Jose
(Deceased)
and
MARJORIE BICKNELL-JOHNSON
State University,
San Jose,
(Submitted
September 1980)
CA 95192
The number of compositions of an integer n in terms of ones and twos [1]
is Fn+1, the (n + l)st Fibonacci number, defined by
F0 = 0 , F1 = 1, and Fn42
= Fn+l
+Fn .
:
Further, the Fibonacci numbers can be used to generate such composition arrays
[2], leading to the sequences A = {an} and 5 = {bn}9 where (an9 bn) is a safe
pair in Wythoff's game [3], [4], [6].
We generalize to the Tribonacci numbers Tn9 where
T0 = 0, Tx = T2 = 1, and Tn+3 = Tn+Z + Tn+1 + Tn.
The Tribonacci numbers give the number of compositions of n in terms of ones,
twos, and threes [5], and when Tribonacci numbers are used to generate a composition array, we find that the sequences A - {An}, B = {Bn}9 and C = {Cn}
arise, where An9 Bn9 and C n are the sequences studied in [7].
1. The Fibonacci Composition Array
To form the Fibonacci composition array, we use the difference of the subscripts of Fibonacci numbers to obtain a listing of the compositions of n in
terms of ones and twos, by using Fn^1, in the rightmost column, and taking the
Fibonacci numbers as placeholders. We index each composition in the order in
which it was written in the array by assigning each to a natural number taken
in order and, further, assign the index k to set A if the kth composition has
a one in the first position, and to set B if the kth composition has a two in
the first position. We illustrate for n - 6,using F7 to write the rightmost
1982]
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
123
column. Notice that every other column in the table is the subscript difference of the two adjacent Fibonacci numbers, and compare with the compositions
of 6 in terms of ones and twos.
FIBONACCI SCHEME TO FORM ARRAY OF COMPOSITIONS OF INTEGERS
INDEX:
A or B
1=
*i
2 - &i
3 = a2
4 - ^3
5 = b2
6 = ah
7 - b3
F6
FB
Fi
Fy
8 = a5
9 = a6
10 = b,
11 = a7
12 = ^8
13 = bs
One first writes the column of 13 F7's9
which is broken into 8 F6fs and 5
F 5 ? s. The 8 F 6 f s are broken into 5 F 5 f s and 3 F 4 ? s 9 and the 5 Fs's are broken
into 3 F 4 f s and 2 F 3 T s. The pattern continues in each column until each F2 is
broken into F± and F0, so ending with F-^. In each new column, one always replaces FnFn's
with Fn_1Fn_11s
and F n _ 2 F n „ 2 ! s. Note that the next level» representing all integers through FQ = 21, would be formed by writing 21 F 8 f s in
the right column, and the present array as the top 13 = F 7 rows, and the array
ending in 8 F6's now in the top 8= F 6 rows would appear in the bottom 8 rows.
Notice further that this scheme puts a one on the right of all compositions of
(n - 1) and a two on the right of all compositions of (n - 2).
Now, we examine sets A and B.
ni
1
2
an:
1
3
bn'.
2
5
7 8 9 10
4
9 11 12 14 16
7 10 13 15 18 20 23 26
3
4
6
5
8
6
Notice that A is characterized as being the set of smallest integers not yet
used, while it appears that bn - an + ^- Indeed, it appears that, for small
values of n, an and bn are the numbers arising as the safe pairs in the solution of Wythoff?s game, where it is known that [2]
(1.1)
an = [ria].
bn = [na 2 ],
where [x]
is the greatest integer in x and a = (1 + /J)/2.
characterize A and B by
Further, we can
124
(1.2)
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
[May
am = 1 + a3F3 + .... + akFk , o^ e {0, 1},
bm = 2 + a!fFlf + ... + a ^ , ai
e {0, 1}.
Any integer n has a unique Fibonacci Zeckendorf representation
(1.3)
n = a2F2 + a 3 ^ 3 + a ^ + ... +
akFk,
where a^ e {0, 1} and o^o^^ = 0 5 or, a representation as a sum of distinct
Fibonacci numbers where no two consecutive Fibonacci numbers may be used. Now
suppose 1 is the smallest term in the Zeckendorf representation of n.
Then n
is in the required form for am.
Suppose that the smallest Fibonacci number
used is Fk , where k is even. Replace Fk by Fk_1 + Fk_2, Fk_2 by Fk_ 3 + Fk_h,
Fk_h by Fk_ 5 + Fk_ 65 ..., until one reaches Fh = F3 + F2, so that we have
smallest term 1, and the required form for am.
Similarly, if 2 is the smallest term in the representation of n, then n is
in the required form for bm.
If the subscript of the smallest Fibonacci number used is odd, then we can replace Fk by Fk_1 + Fk_2, Fk_2 by Fk_3 + Fk_h,
..., just as before, until we reach F5 - Fh + F3, equivalent to ending in a 2
for the form of bm.
Thus A is the set of numbers whose Zeckendorf representation has an evensubscripted smallest term, while elements of B have odd-subscripted smallest
terms. Since the Zeckendorf representation is unique, A and B are disjoint
and cover the set of positive integers. Also, the unique Zeckendorf representation allows us to modify the form to that given for am and bm uniquely,
by rewriting only the smallest term.
Now, we can prove that A and B do indeed contain the safe-pair sequences
from Wythofffs game.
Theorem 1.1: Form the composition array for n in terms of ones and twos, using ^ n + 1 on the right border. Number the compositions in order appearing.
Then, if 1 appears as the first number in the kth composition,
k = am = 1 + a3-F3 + ahF^ + ... + akFk , a* e {0, 1},
and if 2 appears as the first number in the kth
composition,
k = bm = 2 + a 4 F 4 + a5F5 + ... + akFk , o^ e {0, 1},
where (am, bm) is a safe pair in Wythofffs game.
Proof: We have seen this for n = 6 and k = 1, 2, . . ., 13 = F7, and by using subarrays found there, we could illustrate n = 1, 2, 3, 4, and 5. By the
construction of the array, we can build a proof by induction.
Assume we have the compositions of n using ones and twos made by our construction, using Fn+1 in the rightmost column. We put the Fn compositions of
(n - 1) below. (See figure on page 125.)
Take 1 £ j <_ Fn. If j e A, then j + Fn + 1 e A as the compositions starting
with 1 go into A and those starting with 2 go into 5, and addition of Fn + 1
will not affect earlier terms used. Note well that no matter how large the
value of n becomes, the earlier compositions always start with the same number
1 or 2 as they did for the smaller value of n, within the range of the construction. Now, if j is of the form
1982]
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
125
1 + a3F3 + a^Fh + ••• + anFn, j e A,
Q + ^ n +1 = 1 + a 3 F 3 + a ^ +
+ anFn +
Fn+1,
and (j + Fn+1) e A, ai e {0, 1}. Since we know that all the integers from 1
to F
n + i " 2 c a n b e represented with the Fibonacci numbers 1, 2, 3, . .., Fn,
for the numbers through F n + 1 we need only F29 F3, . .., F n . Thus the numbers
1, 2, — , Fn + 2 can be represented using 1, 2, . .., Fn + ±9 and we continue to
build the sets A and B9 having both completeness and uniqueness, recalling [1]
that the number of compositions of n into ones and twos is Fn+1.
Also, notice
that there are Fn_2 elements of B in the first Fn integers and Fn_1 elements
of A in the first F„ integers.
Compositions
of n - 1
•n + l
Compositions of n
r
n+l
Fn + 1 + 1
Compositions of
n - 1
3 + Fn + i-
Fn + 1 + 2
F
J
I <j
<Fn,
-n + i
+F
'x n
n >. 2.
2.
The Tribonacci Composition Array
Normally, the Tribonacci numbers give rise to three sets A, B, C [8]:
A = {An i An = 1 + a 3 T 3 + a ^ + • • • } ,
(2.1)
B = {5 n : Bn = 2 + a 4 T 4 + a5T5 + • • • } ,
C = {<?n : C n = 4 + a 5 T 5 + a 6 T 6 + • • - } ,
where a^ £ {0, 1}. Equivalently, see [7], if Tk is the smallest term appearing in the unique Zeckendorf representation of an integer N9 then
N e A if k E 2 mod 3, N e B if k = 3 mod 3, and N e C if k = 1 mod 3,fc> 3,
where we have suppressed T x ~ 1, but T2 = 1 = A19 and every positive integer
belongs to A9 B9 or C9 where A9 B9 and C are disjoint.
126
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
[May
Also, recall that the compositions of a positive integer n using l's, 2's,
and 3Ts gives rise to the Tribonacci numbers, since Tn+1 gives the number of
such compositions [5].
Now, proceeding as in the Fibonacci case, we write a Tribonacci composition array. We illustrate for T6 = 13 in the rightmost column, which is the
number of compositions of 3 into l f s, 2 T s, and 3's. We put the index of those
compositions which start on the left with a one into set A9 those with a two
into set B9 and those with a three into set C9 and compare with sets A9 B, and
C given in (2.1).
TRIBONACCI SCHEME TO FORM ARRAY OF COMPOSITIONS OF INTEGERS
1
T3
1
2
?3
1
1
Ti
1
We note that thus
is the first positive
of Cirs less than An;
Bn; where An9 Bn9 and
1 =
^5
2 = Bi
3 = A2
1
^i
4 = Ci
1
3
T2
1
2
T5
?i
2
2
T5
5 = A3
6 = B2
Ti
1
3
T5
7 = Ak
T2
1
1
Th
?i
2
1 •
Th
2
T,
1
2
Th
2
3
Th
2
^6
10 = As
11 = C2
1
T3
3
^6
12 = A7
2
T,
3
1
1
T5
1
2
INDEX:
A9 B9 or C
8 = As
9 = B3
2
13 = Bu
far the splitting into sets agrees with these rules: An
integer not yet used; Bn is 2An decreased by the number
and Cn is 2Bn decreased by the number of Ci 's less than
Cn are the elements of sets A9 B9 and C of (2.1).
1 - 2
3
4
5
6
7
Ani
1
3
5
7
8
10
12
Bni
2
6
9
13
15
19
22
C„:
4
11
17
24
28
35
41
ni
We next prove that this constructive array yields the same sets A9B9 and C as
characterized by (2.1) by mathematical induction.
We first study the array we have written, yielding the T6 = 13 compositions of n = 5 using l f s 9 2*s, and 3 f s. We write 13 T6 ' s in the rightmost
column. Then, write the preceding column on the left by dividing 13 T6fs into
7 T 5 T s, 4 T 4 ! s, and 2 r 3 's. In successive columns, replace the 7 T 5 ? s by 4
Tn's,
2 2 V s , and 1 Tl9 and the 4 Th ? s by 2 T 3 ? s, 1 Tl9 and 1 Tl9 then the 2
T
T 3 s by 1 T2 and 1 Tx. Any row that reaches T± stops. Continue until all the
1982]
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
127
rows have reached T1.
Notice that the top left corner of the array, bordered
by 7 T51s on the right, is the array for the T5 = 7 compositions of n - 1 = 4,
and that the middle group bordered on the right by 4'2Vs is the 4 = T 4 compositions of n - 2 = 3, and the bottom group bordered by 2 T 3 ? s on the right is
the 2 = T3 compositions of n - 3 = 2. The successive subscript differences
give the compositions of n using l f s, 2 f s, and 3 ? s.
If we write Tn+1Tn+11s
in the right-hand column, then we will have in the
preceding column the arrays formed from TnTnxs on the right, T n - A - i f s on the
right, and Tn.2Tn-2ls
on the right. All the integers from 1 through Tn+1 will
appear as indices because there are Tn+i compositions of n into l's, 2 ! s, and
3 f s. The subscript differences will give the compositions of n into l's, 2 f s,
and 3*s, and we can make a correspondence between the natural numbers, the
compositions of n, and the representative form of the appropriate set. Those
Tn+i compositions are ordered with indices from the natural numbers. Each
composition whose leftmost digit is one is cast into set A; those whose leftmost digit is two are cast into set B; and those whose leftmost digit is three
are cast into set C. Descending the list, we then call the first A9 Al9 the
second A, A2, . . . , the first B, Si, the second B, B2, and so on. We have now
listed the elements of As B, and C in natural order. Since the representations of An, Bn, and Cn from (2.1) are unique, see [7], and since this expansion is constructively derived from the Zeckendorf representation so that the
largest term used remains intact (by the lexicographic ordering theorem [7]),
every integer m < Tn uses only T2, T3, ..., Tn.1 in the representation, and Tn
can itself be written such that the largest term used is Tn_1.
Let j by any
integer, 1 <_ j <_ Tn . Assume that J can be expressed as in (2.1). Then j ' =
j + Tn+1 will be in the same set as j , since all early terms of j and j ' will
be the same. Further, if the leftmost digit of the jth composition is a, where
1 <_ j <_ Tn , then the leftmost digit of the j'th composition, j ' = j + Tn+1 will
be a, since the leftmost digits are not changed in construction of the array.
V\
il
T
Array
-^rc-2
v.Tr\\.
2
!
*™,v 1
1
'
-Ln
\
\
T„
m
m
-^n+l^n+l
3 +
^-1
j + Tn_x
+ Tn
™n+l ~ Tn + Tn_1
+ Tn
I
&
128
COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS
[May 1982]
Recall that set elements are not characterized by the composition, but only by
its leading 1, 2, or 3. Each number in the jth position in the original gives
rise to one in the (j + Tn+1)st
position in the same set A, B9 or C. Also,
j + Tn + Tn+1 belongs to the same set as j .
References
1.
2.
3.
4.
5.
Alladi Krishnaswami & V. E. Hoggatt, Jr. "Compositions with Ones and Twos."
The Fibonacci
Quarterly
13, no. 3 (1975):233-239.
V. E. Hoggatt, Jr., & Marjorie Bicknell-Johnson. "Representations of Integers in Terms of Greatest Integer Functions and the Golden Section
Ratio." The Fibonacci
Quarterly
17, no. 4 (1979):306-318.
A. F. Horadam. "Wythoff Pairs." The Fibonacci
Quarterly
16, no. 2 (1978):
147-151.
R. Silber. "Wythofffs Nim and Fibonacci Representations." The
Fibonacci
Quarterly
15, no. 1 (1977):85-88.
V. E. Hoggatt, Jr., & D. A. Lind. "Compositions and Fibonacci Numbers."
The Fibonacci
6.
7.
8.
Quarterly
7, no. 3 (1969):253-266.
V. E. Hoggatt, Jr., Marjorie Bicknell-Johnson, & R. Sarsfield. "A Generalization of Wythofffs Game." The Fibonacci
Quarterly
17, no. 3 (1979):
198-211.
V. E. Hoggatt, Jr., & Marjorie Bicknell-Johnson. "Lexicographic Ordering
and Fibonacci Representations." The Fibonacci
Quarterly,
to appear.
V. E. Hoggatt, Jr., & A. P. Hillman. "Nearly Linear Functions." The Fibonacci Quarterly
17, no. 1 (1979):84-89.
*
•
*
•
*
*
*
THE CONGRUENCE
Florida
x» = a (mod m),
WHERE (71, <t> (m)) = 1
M. J. DeLEON
Atlantic
University,
Boca Raton,
(Submitted
January
1980)
FL 33431
Craig M. Cordes [2] and Charles Small [4] proved Theorem 1, a result that
W. Sierpinski [3] proved, using elementary group theoretic considerations, for
n being a prime, and J. H. E. Cohn [1, Theorem 7] proved for n - m. Moreover,
Theorem 1 is implicit in some of the solutions to Problem E2446 in the American Mathematics
Monthly (January 1975).
Throughout this paper, m and n will denote positive integers with m > 1.
Theorem 1: Let n be greater than 1. The congruence xn =. a (mod m) has a solution for every integer a if and only if (n, $(m)) = 1 and m is a product of
distinct primes.
Let a-L, a 2 , . . . , am be a complete residue system modulo m. It follows from
Theorem 1 that a", aj» .••* dm* where n > 1, is a complete residue system modulo m if and only if (n, <j>07?)) = 1'and m is a product of distinct primes.
We shall give a simple proof of Theorem 1 and, in addition, prove the following two related results.
Theorem 2:
I.
The following three conditions are equivalent.
The congruence xn
= a (mod m) has a solution for every integer a with
(a'T^o)=1II.
III.
The congruence xn = a (modm ) has a solution for every integer a relatively prime to m.
(n, cj>(77z)) = 1-
From Theorem 2, it follows that for a19 a2,
..., a<j,(W) a reduced residue
system modulo m9 a", a"» •••> ^J(m) i s a reduced residue system modulo m if
and only if (n, (j)(m)) = 1.
The following result tightens the equivalence of Theorem 2.
Theorem 3:
I.
Conditions I and II are equivalent.
The congruence xn
E a (mod m) has a solution if and only if
(a'T^o)=1II.
(n, cf)(w)) = 1 and pn
+1
j( m for all primes p.
By Theorem 3, we can, with only the simplest of calculations, write down
the nth-power residues modulo 77? if (n, §(m))= 1 and p n+1 j( m for all primes p.
129
130
THE CONGRUENCE x" = a (mod m)9 WHERE (n, $(m)) = 1
[May
We shall now state and prove several results needed for the proofs of
these three theorems.
Lemma 4: Let a and n be positive integers.
T-i™ ^
_ ^ +
.
a positive
integer
t „„„£
such ^
that
Proof: Assume (a, -r-
If ( , -,
r-| = 1, then there is
V
(a, 77Z) /
r-J = 1 and, for convenience, let d = (a, m). Since
fa, ^ j = 1
and
(M ^ j |<|>0w),
by the Euler-Fermat theorem,
a*<n> = 1 (mod | ) . '
There are positive integers a and t such that nt .-. (n, $0??)) = §{m)o.
a»t-(».
Thus
•(».)) = a*(»)» = ! (mod J ) .
Hence
an* = a(n'*(m)) (mod m ) .
Corollary 5: If (n, <f>(m) ) = 1> then the congruence xn
tion for every integer a with (a, -p
= a (mod m) has a solu-
r-) = 1.
Corollary 6: If (ft, §(m) ) = 1 and w is a product of distinct primes, then the
congruence xn = a (mod m) has a solution for every integer a.
Corollary 6 follows directly from Lemma 4 since m being a product of distinct primes implies (a, y
^-y) = 1 for every integer a.
Lemma 7: If the congruence xn = a (modtf?)has a solution for every integer a
relatively prime to m, then (ft, cK^)) = 1.
Proof: Assume (ft, (J)(^)) ^ 1. Thus, there is a prime a such that q\n and
a |(J)(pe) , where p e | |;72 and p is a prime. We shall show that the assumption p =
2 leads to a contradiction and that the assumption p > 2 also leads to a con^
tradiction.
First, assume p = 2. Thus, q divides <j)(2e') = 2 6 " 1 so q = 2 and g J> 2.
Choose a such that a = 3 (mod 2e) and a = 1 (mod m/2e).
Thus (a, /??) = 1; so,
by assumption, the congruence xn = a (mod m) has a solution. Since 4|2e and
2e\m,
we have 4|w. Hence, the congruence ;rn = a = 3 (mod 4) has a solution.
But xn = 3 (mod 4) is impossible, since ft is divisible by q = 2.
Now assume p > 2. Choose a such that a is a primitive root modulo pe and
a = 1 (mod mlpe).
Thus (a5 m) = ls so there is an integer # such that # n = a
e
(mod /??) . Since p \m9 xn = a (mod pe) . For fc = (J)(pe)/q, afe = ^n/c = 1 (mod pe).
1982]
THE CONGRUENCE x" = a (mod m), WHERE (n, <K"0) = 1
131
The last congruence is true because (J)(pe) = qk9 which divides rik.
(mod pe) is impossible, since a is a primitive root modulo pe and
But afe = 1
<,<$)(pe).
0 < k
We shall now prove Theorem lv First, assume that the congruence xn = a
(mod w?) has a solution for every integer a. Thus 0 , 1 , 2 , . .., (77? - 1)
must be incongruent modulo m. Now if there is a prime p such that p2|w then,
since n > 1, we would have the contradiction
0 n E 0 E p ) " (mod m).
Therefore, m must be a product of distinct primes.
By Lemma 7, we have that
(n, (p(m)) = 1.
Conversely, assume (n, (j)(w)) = 1 and m is a product of distinct primes.
By Corollary 6, the congruence xn = a (mod m) has a solution for every integer a.
,
We shall now prove Theorem 2. Since (a, m) = 1 implies (a, -p
r-1 = 1,
II follows from I. The remaining implications—II implies III and III implies
I—follow from Lemma 7 and Corollary 5, respectively.
To prove Theorem 3, we need
Lemma 8: Let a be an integer.
xn
If p n + 1 )(m for all primes p and the congruence
= a (mod m) has a solution, then (a, —,
V
Proof: Assume the congruence xn
prime p such that p\a
Since p\a
e
and p\m9
p |(a, 772). But since p k
^
'
y
=
1-
(a, m) I
= a (mod 77?) has a solution and there is a
—r-. Choose e such that p e | |m; clearly e <_ n.
and p\-(
p\xn;
rl
so pe\xn.
From p e 1772 and pe\xn9
we have that pe\a,
r-, too, we have the contradiction
so
e+1
p \rn.
^ | ( a , 77?)
Finally, we prove Theorem 3. First, assume condition I. Thus, in particular, the congruence xn = a (mod rn) has a solution for every integer a relatively prime to 777. Hence, by Lemma 7, (n, <\>{m)) = 1. To prove that p n+1 1777
for all primes p, assume there is a prime p such that p n + \m. Thus
(p n s w)
= (p n s — ) > P > 1.
Therefore, by condition I, the congruence xn = p n (mod 777) has no solution.
But clearly x = p is a solution to the congruence xn = p n (mod 777) .
The fact that condition II implies condition I follows from Lemma 8 and
Corollary 5.
References
1.
John H. E. Cohn. "On 777-tic Residues Modulo n."
5, no. 4 (1967):305-318.
The Fibonacci
Quarterly
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
132
[May
Craig M. Cordes. "Permutations Mod m in the Form xn." Amer. Math.
Monthly
83 (1976):32-33.
3. W. Sierpinski. "Contribution a 1Tetude des restes cubiques." Ann. Soo.
Polon. Math. 22 (1949) -.269-272 (1950).
4. Charles Small. "Powers Mod n." Math. Mag. 50 (1977):84-86.
2.
•kick-kic
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
CH. A. CHARALAMBIDES
University
of Athens, Panepistemiopolis,
Athens 621, Greece
(Submitted February 1981)
Abstract
The numbers
A{m,
k,
s,
r)
= [Vw+1E*(ga? + r ) J T _ n ,
where V = 1 - E" 1 , EJf(x)
= f(x + j ) , u_x = ux when 0 <_ x <_ k and u_x - 0 otherwise, (y)m - y(y - 1) .•• (y - m + 1), are the subject of this paper. Recurrence relations, generating functions, and certain other properties of these
numbers are obtained. They have many similarities with the Eulerian numbers
A
7
- JLrv m+1 F k r m 1
and give in particular (i) the number Cm,ni8 of compositions of n with exactly
m parts, no one of which is greater than s, (ii) the number Qs,mQ<) of sets
{i x , i2» •••» ^m) with in £{l» 2, ..., s} (repetitions allowed) and showing
exactly k increases between adjacent elements, and (iii) the number Qs>m(r9
k)
of those sets which have i1 = v. Also, they are related to the numbers
G(m9
n, s,
r)
= -^-[kn(sx
+ r)m]x
= Q,
A = E - 1,
used by Gould and Hopper [11] as coefficients in a generalization of the Hermite polynomials, and to the Euler numbers and the tangent-coefficients Tm.
Moreover, lim s~mm\A(m9
k9 s9 su)
= Amt
k
'
S -*• ±oo
A„.k.u
u9
where
*
= ;frrV,+1E*fa + u ) m U . n
is the Dwyer [8, 9] cumulative numbers; in particular,
lim s~mm\A{m9
k9 s) = Am
k9
A(m9 k9 s)
= A(m9 k9 s9 0 ) .
8 +±oo
Finally, some applications in statistics are briefly discussed.
1982]
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1.
133
Introduction
A partition of a positive integer n is a collection of positive integers,
without regard to order, whose sum is equal to n. The corresponding ordered
collections are called "compositions" of n. The integers collected to form a
partition (or composition) are called its "parts" (cf. MacMahon [14, Vol. I,
p. 150] and Riordan [16, p. 124]). The compositions with exactly m parts, no
one of which is greater than s, have generating function
and therefore the number Cm,n, 8 of compositions of n with exactly m parts, no
one of which is greater than s9 is given by the sum
"•••••-& (-,)'(")(" " . - i * 0 -
"•»
where k = [(n - m)Is], the integral part of (n - m)/s.
Compositions of this type arose in the following Montmort-Moivre problem
(cf. Jordan [12, p. 140] and [13, p. 449]): Consider m urns each with s balls
bearing the numbers 1, 2, ..., s. Suppose that one ball is drawn from each
urn and let
m
z = E *i
be the sum of the selected numbers. Then the probability p(n; m, s) that Z is
equal to n is given by
(1.2)
p(n; m, s) = s~mCms n> s, n = m9 m + 1, ..., sm.
Carlitz, Roselle, and Scoville [4] proved that the number QsymOO* of sets
{£ l5 i2> »''3 ^m} with in z {l, 2, ..., s} (repetitions allowed) and showing
exactly k increases between adjacent elements, is given by
Q..mW-£l-i)<(m+1l)(B<*-*l+m-1).
d.3)
and the number QSj m(r9
d.4)
k) of those sets which have i,1 - v is given by
0...(r. k) = *E (-I)'(J )(S(fc " J' " i'.V + " " ')•
The next problem is from applied statistics: Dwyer [8,9] studied the problem of computing the ordinary moments of a frequency distribution with the use
of the cumulative totals and certain sequences of numbers. These numbers are
the coefficients ^ a . r of the expansion of (x + r)m into a series of factorials (x + k)m, k = 0, 1, 2, ...,77?; that is,
m
(* + ^
= E
k=o
4m,fe,pto+.7W -
k)mlm\
Using the notation u_x = ux with 0 <_ x <_ k and u_x - 0 otherwise, he proved that
,,.
•
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
fM
imy
k
. A m , k , r = [V^E^Qr + r) m ] r _ 0 = Z
£ (-l)J(m + M(fc - .j +r)"..
(1.5)
These numbers for r = 0 reduce to the Eulerian numbers
A n . k - [V" +1 EV]«.o
(1.6)
=
E (-^'(^ * X ) ^ " ^m'
x
j-o.
J
'
In t h e present p a p e r 9 starting from the problem of computing the factorial
moments of a frequency distribution w i t h the u s e of cumulative t o t a l s , w e i n troduce the numbers
(1.7)
A(jn, k,
r) = ±-[Vm+1Ek(sx
s,
+ r).].. n
so that
m
= 2 ^( m > &» *»• i»)(a: + m - fe).m.
fe = o
T h e s e numbers h a v e many similarities w i t h the Eulerian numbers (cf, Carlitz
[ 1 ] ) . They a r e related to the numbers Cmtn>8
of the title of this paper b y
(1.8)
Cm.n.a
(s# + r)m
= A(m - 1, k - 1, 8, v + m - 1) = (-ir--\A(m - 1,fc- 1, - s , - r - 1 ) ,
fc = [ ( n - m ) / s ] ,
3? = (n - w) - s[ (n - w ) / s ] ,
and t o t h e numbers Q8tm(^9 k) by
Q8tm(r,
k) = i4{m - 1, fc - 1, s , r + m - 2) = ( - l ) 7 " " 1 ^ ^ - 1, .fc - 1, - s , - r ) ,
and t h e i r p r o p e r t i e s a r e d i s c u s s e d i n S e c t i o n s 2 and 3 below.
i t follows
Since
that
•Sa^tfO = 4 ( ^ k - 1 , 8 , s + m - 1) = ( - l ) M ( m , k ; - 8 ) ,
w h e r e i4(m, /c, s ) = ,4(777, k9 s , 0 ) . Section 4 is devoted to the discussion of
certain statistical applications of the numbers A(m9 k9 s , r ) .
2.
Let {x)mtb
The Composition Numbers A[m9 k, s,
r)
- x(x - b) . .. (x - mb + b) denote the generalized falling fac-
torial of degree m w i t h increment b;
the usual falling factorial of degree m
w i l l b e denoted by ( x ) m = 0*0 m,i'
T h e Problem of expressing the generalized
factorial { x ) m , b in terms of the generalized factorials (x + ka)m,a9
k = 09 1,
2, . . . , m of the same degree arises in statistics in connection w i t h t h e p r o b lem of expressing the generalized factorial m o m e n t s in terms of the cumulations (see Dwyer [8, 9] and Section 4 b e l o w ) . M o r e generally, let
m
(2.1)
(x + rb)nub
= £ Cm, k, r(a9
k=>o
b){x
+ (m -
k)a)m>a.
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1982]
135
Following Dwyer, define u^ = ux when 0 <_ x <_ k and u_x = 0 otherwise. Moreover, let E a denote the displacement operator defined by Eafix) = fix + a) and
Va = 1 - Ea 1 , the receding difference operator; when; a = 1, we write E r =• E
and Vx E V. Then, from (2.1), we have
(2.2)
(* + rfr) m ^ - I C » . t , , ( a , i ) ( a + (w'-fe)g^ 3.
&=0
'
Since
\amml,
[Vm+1E2(x + (m - W a ) B , f l ] 8 . 0 =1
k = n
?
I 0 , 0 <_ fc < n
o r n < k <_ m,
we g e t , from ( 2 . 2 )
rr~m
cm, k, , ( a , w = fr[v a
m+1
k
t
E a (£^Lz*) m> 6 ]«-o-
These coefficients may be expressed in terms of the operators V and E and the
usual falling factorials by using the relations
V a " +x E*/0B) = Vm+1Ekf(ax),
(ax + vb)m,b
= b™ (ax + x>)„, s = alb.
We find
c
m,k, r(a> b) = s~mA{m,
where
(2.3)
Aim, k, s, r) = -~-[\Jm+1Ekjsx
k, s , r ) , s = alb,
+ r ) X . n / i c = 0, 1, . . . , w,
" m = 0, 1, 2, . . . .
Hence
m
(2.4)
fo
+ rb)m
fc
or
= 2 s' m 4(77z, fc, s , r ) ( x + (m - k)a)m,a9
fe = o
s = a/2?
m
(2.5)
(arc + r)m = 2 ^C777' &> s » r ) ( f e + 7 7 7 - fc)w.
fc = 0
Using the symbolic formula
we get, for the numbers (2.3), the explicit expression
(2.6)
A(jn,k,s,r)
=' ]£<-»< (™ 1%'*
r
f
+
%
It is easily seen that
(2.7)
Aim, k, -s,
-r) = i-l)mAim,
k, s , r+m-
1),
and, also, that the numbers Aim, k, s, r) are integers when s and r are integers. Moreover, Aim, k, s, r) = 0 when k > m.
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
136
[May
Remarks 2.1: As we have noted in the introduction, the number CmtntS of compositions of n with exactly m parts, none of which is greater than s is given
by
(2-8)
Cmtn>s
= E <-!>'(")(" ~m"f i
Comparing (2.8) with (2.6) and using (2.1),
(2.9)
A(m - 1, k9
CmtniS=
= {-l)m'1A{m
* ) . k = [<n
-m)/s].
we get the relation
s , v + m - 1)
- 1, k,
-s9
-v
- 1 ) , r = (n - m) - s[(n
-
m)/s],
which justifies the title of this section.
Since the number Qs,m('^9 k), of sets {il9 i2* •••> ^m) with in e {l, 2,
..., s} (repetitions allowed) and showing exactly k increases between adjacent
elements which have ix = p, is given by (see [4])
(2.io)
Q.,m<r,
k) = jE<-i)'(S)( a ( * " 1 ~ f _V + m ' %
we get, by virtue of (2.6), the relation
(2.11)
Qs,m(r,
k) = A(jn - 1, k - 1, s, r + m - 2)
= (-l)ffl"1A(wi - 1, k - 1, -8,
-r).
These numbers give in particular the numbers
k
Qs,m(k) = JEC-U'f" .J ' X 3 ^ ~ ' V ™ ~ *)
(2.12)
of the above sets without the restriction i
i1x = r.
,mQQ.
\We have
= e s , m+] _(s, k>»
and hence
(2.13)
efl,m(k) = i*(m,-fc- 1, s, s + m - 1) = (-l)mA(m9
k9 - s ) ,
A(m, k, s) = ^-[V" + 1 E f c ( e a r) m ] x . 0 = £ (-l)'(m +. l){s{k
~
where
(2.14)
J)
).
Since
it follows that
:
+
+
£ <-< j )r -« ') - .;?>•>'"(•; or v 1
and (2.6) may be rewritten as follows:
0 NT H E
iQooi
ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
J
137
m+ l
A(m, k, s, r) = £ (-iy'+ 1 ( m + l)(8^k ~ 3) + A
_ "y* f_if+itm
+ l\/-s(m -fe+ 1 - i) + r\
= m'y1(-l)i(m
+ 1
)(8^"
- k + l - i ) + m - r - l \
Hence
A(w,fe,s, r) = i4(m, m - k + 1, s, wz - r - 1) + O - ^ L !! £ + 'l)(m)
= i-D'AOn. » - * + ! . -s, *) + (-1)^ » + * ^
.
In particular
A(m9 k9 s) = (-l)mA(m9
(2.16)
m - k + 1, -s) ,
which should be compared with the symmetric property of the Eulerian numbers
"•m, k
=
"-m, m- k + \ •
Using the relation
(m + 2)(s(k
- j) + v - m) = (sk - m + r)(mt ^ - (8(m - k + 2) + m - r)(™ + J),
we get9 from (2.6), the recurrence relation
(2.17)
(m + 1)4 (m + 1, k9 s9 r)
= (sk - m + r)A(m9 k9 s9 r) + {s(m - k + 1) + m - r)A(m9 k - 1, s, r)
with initial conditions
4(0, 0, s, r) = 1, A(m9 0, s, r) = (*j,
TW.>
0.
From (2.5), we have
m
(s(k - j) + r)m = £ 4 0??, m - i9 k - j, r) (s + £)OT.
i=0
Hence (2.6) may be rewritten as
k
A(rn,k,s,r)--i(-iy(my)(S«-f+r)
£ (8+ml)Mm,m - i, k-j, r)
= i^fV)
- f(8!i)Z:(-lW'nt W m - i ,fc-j.r).
i = 0X
A
"
'J=0
X
^
'
0N T H E
m
'
ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
fM
U
' ay
and putting
k
(2.18)
B(m,
n, k9. .r)
= £
(-lW™
t
1
k - j9
)A,(PI9 n,
P)
we get
(2.19)
Mm9 k9 s, r>,- E T * * W m , W - i9 k9 P) ,
i = o \ r/n I
and
(2.20)
B(m, n, k, r) = £
£
(-l)i + '(m + ^
+ l)^n
" *>» " ^
+
*).
It is clear from (2.20) that
(2.21)
B{m, n9 k9 P ) = B(m9 k9 n9 p ) .
Since
2 m
{""I
)( y)((n-i)(k-j)+r-m)
- <n* - m + r)(™ J ^
+ X) -
(»QH
- (fcOn - „ + 2) + m - r)(m.
- * + .2) + m - r)( m t ; 1 ) ^ + }j
+
_ ^)(m
+ X
. )
+ ( < * - « + 2) On -fc+ 2) - m + r)(™ t j)(j ! J).
it follows, from (2.20), that
(2.22) {m + l)B(m + 1, n,fc,P )
= (n/c - m + r)B(m9
n,fe,p) + (n{m - k + 2) + m - r)B(m9
'"+ (fc(m - n'+ 2) + m - v)B(jn9
n, 7c - 1, P )
n - 1,ft,P )
+ ((m - n + 2)(/H - k + 2) - m + r)B(m9
n - 1,fc- 1, P ) .
with
B(0, 0, 0, p) == 1, £(0, 0,ft,P ) = £(0,ft,0,
P)
= 0.
Remark 2.2: Comparing (2.20) with the formula
giving the number of permutations on m letters which have n jumps and require
k readings (cf. [4]), we find
(2.23)
Rm(n9 ft) = B'(m9 n,ft,m - 1) = B(m9 m - n + 1, k)
- B(rn9 n9 m - k + 1),
where
(2.24) B(m, n, *) E B(m, ,, 'fc, 0) = &
t ^ l f * ^ *
1
) ^ J * )( ( W " *><* " *>) •
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1982]
139
Using the relation
j : !.«-»'*'(" I'X" J 1)(0,"*>.<*"i0)rf.r+ii^r+i
}
\ i
A 3 ) \
m
)'
it can be easily shown that
(2.25)
B(m, n, k)=B(m,
m - n + 1, m-
k + 1).
Expanding the generalized factorial (x+ rb)m,b
in terms of the generalized
factorials (x+ka}m,a9
k = 0 9 1, 2, .. ., m and then these factorials in terms
of the factorials (x+jb)m,b,
j = 0, 1, 2, . . . , m, by using (2.4), we get
m
(x + rf>)m>ib = J ] a'mbmA(m9 m - k, alb, r) (x + ka) m , a
fc«0
mm
= E E 4 .On, m -'fc, a/fc, I'Mlm, w? - j , i / a , &).'(# + jb)m
b
k - 0 j *. 0
m m
= £
H ^ O " , m - fe, alb,
j-0
which implies
(2.26)
^ ^.(w> w ~ &» a/fr* FMOW, m - j , b/a,
fe-o
with 6rj the Kronecker delta:
pair of inverse relations
(2.27)
v)A{m, m - j , b/a,
r)(x
+ jfc)
.,
fc-0
ar = Yl
A m
6rr
( > m - k, a/bs
r) = 6 r Ji -,
= 1, 6rj- = 0, j ^ r.
r)$k,
Hence, we have the
$k = J2 A(m, m - k, b/a,
r)ar.
3. Generating Functions and Connection with
Other Sequences of Numbers
Consider first the generating function
(3.1)
Am,8,r(t)
- 2X^>
k9 s,
r)tk,
where the summation is over all possible values of k which are 0 to m and can
be left indefinite because A(m, k, s, v) is zero elsewhere. Then, from (2.6),
it follows that
(3.2)
A™,«,,<*) - (1 - t)m+1
i(sk
+
r
)tK
In a generalization of the Hermite polynomials, Gould and Hopper [ 11 ] used
as coefficients the numbers
(3.3)
G(m, n, s, r) = - ^ £ (-1)"' { (n\
(sj + r ) m ,
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1An
i4U
rM
LNay
which may be equivalently defined by
G(m9 n9 s,
v) = -^j[t\n (sx + r)m]x
= 0.
Using the symbolic formula
n-=00 W
k
and since [E (sx
+ r)m]g;mQ
= (sk + v)m9
we get
m
(sk + r)m = £ £(^> n> s,
(3.4)
v)(k)n.
n=0
The generating function (3.2) may then be rewritten as
j
m
^ ( m , n, s, r)tra(l - t)"-",
Am,..,(*) = £
n = 0 "<•
so that
(3.5)
A(m9 k, s, P ) = E o (-l)*" n jf(^ I ^ ( m , " n , s, P ) .
Since for r = 0 the numbers G(m9 n9 s9 r) reduce to the numbers
-^j[kn(sx)m]x=0
C(jn9 n9 s) =
studied by the author [5, 6, 7] and also by Carlitz [2] as degenerate Stirling
numbers, we have, in particular,
A(m9 k9 s) = £ (-Dk"nSf(!J I k)C(m>
(3.6)
n
> S)'
The generating functions
(3.7)
Ae,r{t,
x)
= £
E
m±Q
A(jn, k,
s,
r)tkxm
k=0
and
(3.8)
As(t9
x9 y) = £
£
r)tkyrxm9
f ^ f a , fc, s ,
m= 0 r = 0 & = o
using (3.2), may be obtained as
(3.9)
*aip(*. .) - d - t ) [ i
( i - t )
+
g
r
a
1 - £[1 + (1 - t)x] s
(3.10)
As(t9
(1
x9 y) =
{1 - t[l
Since 11m A8t r(t9
(3.11)
x)
Am>3i
s
" *}
+ (1 - t)x] }{l
.
- y[l
= (1 - s x ) " 1 , we g e t
,(1) = £ i ( n ,
fc = o
&, s , r ) = s m .
+ (1 - t ) x ] }
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1982]
141
Using (2.19) and (2.21), (3.11) may be rewritten in the form
m
m + 1i
.
sm = E
Er
'
fc = 0 i = l \
m+1
m
L
.
J
v
>(^ 5 i, m - fc +1, p)
m
k=0 i =l \
v)
/
- E E r ' '
= E
)B(m, m - i + 1, k,
"l
/
„"
E B(m, i, m - k + 1, v).
It is known that the Eulerian numbers Am i satisfy the relation (see [19]
or [1])
£(,+i-'K
Therefore
m+i
(3.12)
.
The g e n e r a t i n g
^
B f a , £ , fc, r)
= i4Wii.
function
m
£ m s n > 2 , ( £ ) = E B ( m , n , fc, p ) t k
(3.13)
fc = o
is connected with A m> Sj r (t) by the relations
(3-14)
s+
E T( „ "JB,,.
^ „.<.,(*)
-E
Am,Str(t)
and
(3.15)
Bn, „,,(*) = Z X - D ' f " } V , , . ^ ^ * ) .
\
j-o
Returning to (2.6), let us put r = sw.
(3.16)
«/
/
Then
l i m s""mM(/n, fc, s , r ) = 4 m , &,«,
S -*• ±oo
where
4,.*.« - [V m+1 E fe fa+u)-]_ n = E(-D^'( m t
X
W +u - j)"
are the numbers used by Dwyer [8] for computing the ordinary moments of a frequency distribution. In particulars
lim s-mm\A(m9
(3.17)
s) = Am
k,
k.
'
S -* ± 0 0
Consider the function
(3.18)
Hm(t;
s, r)
= (1 - * ) " X , «. P (*)'
= E
Then, using (3.3), we get
(3.19)
H(x;
i, s9 r)
nsQ
£rff(m,.n, s, r)t"(l - t ) " n .
m.
= E # m ( * ; s, a?)** = (1 - t) (1 + x)r[l - t(l + a?)*]""1.
m=0
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
142
Since lim H(x/s;
[May
-1, s9 su) = E(x; u), where
8 -*• ± <»
E(x; u) = Y, Em(u)xm = 2e*w/(l + ex)
(3.20)
m- 0
is the generating function of the Euler polynomials ([12, p. 309]), it follows
that for the polynomials
m
Cm(u; s) = ffm(-l; s, su) = 2"m 2 (-l)k4fa> fc, 8, su)
£=0
we have
lim s"mt>m(u9
s) =
Em(u),
S -*- ± oo
which, on using (3.16), gives
m
(3.21)
£ (-D^m.fc.u -
2mm\Em(u)
k =0
and, in particular,
m
E(-Dfc^.fc.i/2-^.
&=o
(3.22)
where Em = 2mm\Em(l/2)
is the Euler number ([12, p. 300]).
Putting u = 0 in (3.21), we get
m
£(-l)"Vfc = 2'^
(3-23)
fc = 0
where T m = 2mm\Em{0)
is the tangent-coefficient ([12, p. 298]).
Remark 3.1: The degenerate Eulerian numbers Amtk{\)
3] by their generating function
(3.24)
i + £
fr £4m.k<x)t* = —
m-1 m ' k-1
are related to the numbers
A(m9
introduced by Carlitz [2,
-—
1 - £[1 + Xx(l -
—
t)]1/X
k9 s) = i4(77?, Zc, s, 0 ) .
Indeed, comparing (3.24) with (3.9), we get
A(m9
4.
k, s) = —
Amtk{s-1).
Applications in Statistics
The numbers A(m, k9 s9 r) like the Eulerian numbers Am,-k seem to have many
applications in combinatorics and statistics. Special cases of these numbers
have already occurred in certain combinatorial problems, as was noted in the
introduction. In this section, we briefly discuss three applications in statistics. The first is in the computation of the factorial moments of a frequency distribution with the use of cumulative totals. This method was suggested by Dwyer [8, 9] for the computation of the ordinary moments, as an
alternative to the usual elementary method and, therefore, for details, the
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1982]
143
reader is referred to this work. We only note that the main advantage of this
method is that the many multiplications involved in the usual process are replaced by continued addition. Let fx denote the frequency distribution and
Cm + 1fx = C(Cmfx)9
m = 1, 2, 3, ..., Cfx
^hfj*
the successive frequency cumulations. Then, from the successive cumulation
theorem of Dwyer, we get9 for the factorial moments,
k
(4.1)
m
£ (ax + r)mfsx+r
= £ m\A(jn, n, s,
x-0
v)Cm+1fsn+I,.
n=0
When r = 0, i.e., when the factorial moments are measured about the smallest
variate, (4.1) reduces to
k
(4.2)
m
Z^x)mfsx=
T,m\A{m,
rc= 0
s)Cm+1fsn,
n,
n=0
which for s = 1, i.e., when the distance between successive variates (class
marks) is unity, gives ([8, §9])
k
(4.3)
m
= £m!il<m, n, l)Cm+1fn
Z&)mfx
x=0
-mlC»i\,
n=0
since A(m9 m9 1) = 1, A(m9 n , l ) = 0 if n # m.
The second statistical application of the numbers A(m9 k9 s9 r) is in the
following problem: Let X1, X2, ..., Xm be a random sample (that is, m independent and identically distributed random variables) from a population with
a discrete uniform distribution
p(n;
s) = P(X = n) = s" 1 , n = 0, 1, 2, ..., s - 1.
Then the probability function of the sum Zm = X xi may be obtained as
i =l
(4.4)
= « - £ (-1)* (j )(» + ^
p(n;m>S)
= s"mA(m
} ~ 8 ')
- 1, [n/s], s, P + m - 1),
n = sk + P ,
0 <. r < s.
Note that the distribution function
F
[w]
2 pWs;
=
m 0fa)
n=0
m9 s)
of the sum
m
Wm=Y*Yi9Yi=s-1Xi9i
=
l,29...,m
i =l
approaches, for s -> oo, the distribution function
[u]
Fm(u). x 2(-i)^(J)( M - ' j r
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1ZLA
144
rM
LNay
of the sum
m
i =l
of m independent continuous uniform random variables on [05 1) (Feller
and Tanny [18]).
Since
E(Zm) = mE(X) = m(s - l)/2,
Var(Zm) = wVar(J) = m(s2
[10]
- 1)/125
it follows from the central limit theorem (see, e.g., Feller [10]) that the
sequence
Zm - m{s - l)/2
/m(s2
- 1)/12
converges in distribution to the standard normal.
lim
mJhto
(4.5)
V
s'mA(m
- 1, k9 s9 r)
Hence
= $(g),
k=o
s w = z/m(s2
- 1)/12 + m(s - l)/2.
and
(4.6)
lim /w(s 2 - l)/12s-m4(/?2 - 1, [zm] , s, r + m - 1) = <p(z) ,
m->oo
where <^(s) and <£>(s) are the density and the cumulative distribution functions
of the standard normal.
Finally, consider a random variable X with the logarithmic series distribution
p(k;
0) = P(X = k) = aQk/k9
k = 1, 2, ..., a"1 = -log(l - 0 ) , 0 < 0 < 1.
Patil and Wani[15] proved the following property of the moments \im(0) = E(Xm) :
ym(6) = a(l - QymY,c(m
- 2, fc)9k+1,
fc = o
where the coefficients satisfy the recurrence relation
c(m9 k) = (Zc + l)c(m - 1, Zc) + (TH -fc+ l)e(w - 1,fc- 1),
c(0, 0) = 1, <?(m, k) = 09 k > m.
It is not difficult to see that
e(m, fc) =
w i t h t h e l a t t e r a E u l e r i a n number.
y n ( 0 ) = a ( l - 0)~mtAm-i
fc = 0
Am+Uk+1
Hence
k
k®
=
a
^
"
^ " X - i W -
A similar result can be obtained for the generalized factorial moments
ON THE ENUMERATION OF CERTAIN COMPOSITIONS
AND RELATED SEQUENCES OF NUMBERS
1982]
y(IB.fc)(e)
in terms of the numbers AQn, k5 s, r) .
145
=E[(x)mtb]
Indeed , we have
y<»;i) (©) = a E(fe) W f i e f c /fe = £ £ (fc - i ) W i J
as
-m+1
2 (*fc - D^e*, * = zr1,
fc-l
and s i n c e , by ( 2 . 1 7 ) and ( 2 . 1 8 ) ,
00
tTmAm_ltStV(t),
£ ( s f c + P ) ^ * * = (/n - 1 ) 1 ( 1 k-l
i t follows
(4-7)
that
y ( n , . 6 ) (6) = a s - r a + 1 ( l - BYm (m - 1) lAm_lt
which, i n p a r t i c u l a r ,
y ( m . D (9)
s>
., (6),
gives
= a(l
- e)"w(m -
1 ) ! 5>(TTZ -
1. &» 1,
-1)9*
k= l
= a(m - 1) 10^(1 - Qym .
References
1.
L. Carlitz. "Eulerian Numbers and Polynomials." Math. Mag. 32 (1959):
247-260.
2. L. Carlitz. "Degenerate Stirling, Bernoulli and Eulerian Numbers."
Utilitas
Mathematica
15 (1979):51-88.
3. L. Carlitz. "Some Remarks on the Eulerian Function." Univ. Beograd.
Publ.
Elektrothen.
Fak. 3 Ser. Mat. Fiz. 602-633 (1978):79-91.
4. L. Carlitz, D. P. Roselle, & R. A. Scoville. "Permutations and Sequences
with Repetitions by Number of Increases." J. Comb. Theory, Series A (1966):
350-374.
5. Ch. A. Charalambides. "A New Kind of Numbers Appearing in the n-Fold Convolution of Truncated Binomial and Negative Binomial Distributions." SI AM
J. Appl. Math. 33 (1977):279-288.
6. Ch. A. Charalambides. "Some Properties and Applications of the Differences
of the Generalized Factorials." SIAM J. Appl. Math. 36 (1979):273-280.
7. Ch. A. Charalambides. "The Asymptotic Normality of Certain Combinatorial
Distributions." Ann. Inst.
Statist.
Math. 28 (1976):499-506.
8. P. S. Dwyer. "The Calculation of Moments with the Use of Cumulative Totals." Ann. Math. Statist.
9 (1938):288-304.
9. P. S. Dwyer. "The Cumulative Numbers and Their Polynomials. Ann.
Math.
Statist.
11 (1940):66-71.
10. W. Feller. An Introduction
to Probability
Theory and Its
Applications.
2nd. ed. Vol. II. New York: John Wiley & Sons, 1971.
11. H. W. Gould & A. T. Hopper. "Operational Formulas Connected with Two Generalizations of the Hermite Polynomials." Duke Math. J. 29 (1962):51-63.
12. Ch. Jordan. Calculus
of Finite
Differences.
2nd ed. New York: Chelsea,
1960.
146
A GENERALIZATION OF THE GOLDEN SECTION
[May
13. K. Jordan. Chapters on the Classical
Calculus
of Probability.
Akademiai
Kiadb, Budapest, 1972.
14. P. A. MacMahon. Combinatory Analysis,
Vols. I and II. New York: Chelsea, 1960.
15. G. P. Patil & J. K. Wani. "On Certain Structural Properties of the Logarithmic Series Distribution and the First Type Stirling Distribution."
Sankhya,
Series A, 27 (1965):271-180.
16. J. Riordan. An Introduction
to Combinatorial
Analysis.
New York: John
Wiley & Sons, 1958.
17. J. Riordan. Combinatorial
Identities.
New York: John Wiley & Sons, 1968.
18. S. Tanny. "A Probabilistic Interpretation of the Eulerian Numbers." Duke
Math. J. 40 (1973):717-722; correction, ibid.
41 (1974):689.
19. J. Worpitzky. "Studien liber die Bernoullischen und Eulershen Zahlen." J.
Reine Angew. Math. 94 (1883):203-232.
ick -kick
A GENERALIZATION OF THE GOLDEN SECTION
University
D. H. FOWLER
of Warwick, Coventry,
England
(Submitted
February
1981)
Introduction
It may surprise some people to find that the name "golden section," or,
more precisely, goldener
Schnitt,
for the division of a line AB at a point C
such that AB • CB =-AC2, seems to appear in print for the first time in 1835 in
the book Die reine Elementar-Mathematik
by Martin Ohm, the younger brother of
the physicist Georg Simon Ohm. By 1849, it had reached the title of a book:
Der allgemeine
goldene
Schnitt
und sein
Zusammenhang mit der
harminischen
Theilung
by A. Wiegang. The first use in English appears to have been in the
ninth edition of the Encyclopaedia
Britannica
(1875), in an article on Aesthetics by James Sully, in which he refers to the "interesting experimental enquiry . . . instituted by Fechner into the alleged superiority of Tthe golden
section' as a visible proportion. Zeising, the author of this theory, asserts
that the most pleasing division of a line, say in a cross, is the golden section . . . ." The first English use in a purely mathematical context appears
to be in G. Chrystal's Introduction
to Algebra
(1898).
The question of when the name first appeared, in any language, was raised
by G. Sarton [11] in 1951, who specifically asked if any medieval references
are known. The Oxford English Dictionary
extends Sarton1s list of names and
references and, by implication, answers this question in the negative. (The
1933 edition of the OED is a reissue of the New English Dictionary,
which appeared in parts between 1897 and 1928, together with a Supplement. The main
dictionary entry "Golden," in a volume which appeared in 1900, makes no reference to the golden section, though it does cite mathematical references that
will be noted later; the entry "Section" (1910) contains a reference to "medial section" (Leslie, Elementary
Geometry and Plane Trigonometry,
fourth edition, 1820) and to Chrystal's use of "golden section" noted above. The 1933
1982]
A GENERALIZATION OF THE GOLDEN SECTION
147
Supplement does not appear to contain any further references. A further Supplement, which started publication in 1972, has a long and detailed entry under "Golden" which is clearly based on and extends, but does not answer, Sarton's question.) Among the other names are: the Italian divina
proportione
(Luca Pacioli, in his book of that name, published in Venice in 1509) or Latin
proportio
divina
(in a letter from Johannes Kepler to Joachim Tanck on May 12,
1608; then in Keplerfs book De Nive Sexangula,
1611); the golden medial; the
medial section; and the golden mean. This last term "golden mean" is credited
by the OED to DTArcy W. Thompson. (Further complications! The OED—1972 Supplement entry "Golden"—cites p. 643 of On Growth and Form [12]: "This celebrated series, which . . . is closely connected with the Seotio
aurea or Golden Mean, is commonly called the Fibonacci series." The reference is to the
now rare first edition of 1917; the second edition has an expanded and elaborately erudite version of this footnote on pp. 923 and 924, which starts differently: "This celebrated series corresponds to the continued fraction 1 +
1/1+ 1/1+ etc., [though Thompson, who uses a slightly different layout of the
fraction, omits the first term in both versions of the footnote] and converges
to 1.618..., the numerical equivalent of the seotio
divina,
or 'Golden Mean.'"
This same dictionary entry later assigns the first use of the Latinized
seotio
aurea to J. Helemes, in 1844, in a heading in the Arohiv fur Mathematik
und
Physik,
IV, 15: uEine . . . Au fid sung der seotio
aurea.11)
Unfortunately, the
same expression "golden mean" is usually applied to the Aristotelian principle
of moderation: avoid extremes. Other quite different things with similar names
are the golden rule (the rule of three; see the OED 1933 edition entry "Golden" for references) and the golden number (the astronomical index of Meton's
lunar cycle of nineteen years). Also E. T. Bell, in "The Golden and Platinum
Proportions" [2], refers to "the so-called golden proportion 6:9::8:12," but I
cannot decide whether this article is meant as a serious contribution or not.
If confusion and misapprehension were confined to nomenclature, that would, it
is evident, be bad enough; alas, more is to be described, after a paragraph of
sanity.
The mathematical theory of the golden section can be found in many places.
I would cite' Chapter 11 of H. S. M. Coxeter's Introduction
to Geometry [4] as
both the best and most accessible reference, and further developments can be
found in other of Coxeter's works. The briefest acquaintance with any treatment of the Fibonacci series will indicate why many accounts of that topic
will tend to the golden section, and The Fibonacci^Q^arterly
is a rich source
of articles and references on this subject. That there appears to be a connection between the Fibonacci numbers (and hence the golden section) and phyllotaxis (i.e., the arrangement of leaves on a stem, scales on a pine cone,
florets on a sunflower, infloresences on a cauliflower, etc.) is an old and
tantalizing observation. The subject is introduced in Coxeter [4], a brief
historical survey is included in a comprehensive paper by Adler [1], and Coxeter [5] gives a short and authoritative statement.
The application of the golden section to other fields has, however, created a vast and generally romantic or unreliable literature. For instance,
the application to aesthetics is, by its nature, subjective and controversial;
a good brief survey with references is given in Wittkower [ 13] . For a comprehensive example of the genre, see the rival explanation and critical view of
the role of the golden section in literature, art, and architecture in Brunes,
The Seorets
of Ancient
Geometry [3]. (Lest I be incorrectly understood to be
dismissing the scientific and experimental study of aesthetics as worthless,
let me cite H. L. F. von Helmhlotz's On the Sensations
of Tone [10] as an
148
A GENERALIZATION OF THE GOLDEN SECTION
[May
impressively successful example of this type of investigation, the very acme
of science, mathematics, scholarship, and sensibility. In particular this book
contains the first explanation of the ancient Greek observation that harmony
seems to be connected with small integral ratios. But it is precisely Helmholtzfs masterly blend of acoustics, physiology, physics, and mathematics that
establishes firmly a standard which so few other writers on scientific aesthetice approach.)
With this outline of the recent history of the golden section behind us,
my objective here is to treat the construction as it is described in Euclidfs
Elements under the name of "the line divided in extreme and mean ratio" and to
develop and explore beyond the propositions we find proved there. My covert
purpose is historical: to pose implicitly the question of whether the generalizations to be described here might have had any part, now lost, in the development of early Greek mathematics. To isolate this discussion of the ancient
period from the later convoluted ramifications sketched in this introduction,
I would like to finish with what is, I hope, an accurate description of the
surviving evidence about the Greek period: the propositions to be found in
Euclid's Elements
constitute the only direct, explicit, and unambiguous surviving references to the construction in early Greek mathematics, philosophy,
and literature; and the only other surviving Greek references are to be found
in mathematical contexts, in Ptolemy's Syntaxis^
Pappus ' Colleotio,
Hypsicles1
"Book XIV" of the Elements3
and an anonymous Scholion on Book II of the Ele-
ments.
Acknowledgments
I would like to thank I. Adler, M. Brown, H. Cherniss, H. S. M. Coxeter,
R. Fischler, S. Fowler, T. 0. Hawkes, S. J. Patterson, E. Shiels, D. T. Whiteside, C. Wilson, and many members of Marlboro College, Vermont, for diverse
kinds of help during the preparation of this article.
The Definition in Euclid's Elements
The golden ratio is defined at the beginning of Book VI of the
Elements'.
A straight line is said to have been cut in extreme and mean ratio
when, as the whole line is to the greater segment, so is the greater
to the less.
Book VI applies the abstract proportion theory of Book V to geometrical magnitudes, and Proposition 16 describes how to manipulate the proportion in the
definition above into a geometrical statement:
If four straight lines be proportional, the rectangle contained by
the extremes is equal to the rectangle contained by the means; and
if the rectangle contained by the extremes is equal to the rectangle
contained by the means, the four straight lines will be proportional.
Otherwise said, if a, b, c, and d are four lines such that aibiia:d9
then rectangle (a, d) = rectangle (£>, o) and conversely. Hence, if C divides the line
AB in the golden section, the rectangle with sides AB and BC is equal to the
square with side AC.
1982]
A GENERALIZATION OF THE GOLDEN SECTION
149
This is meant literally. An elaborate
theory9 now generally called the "application of areas," is developed in the Elements,
and this describes how, for example, to
manipulate any rectilinear plane area into
another area equal to the original area and
similar to a third figure. Our arithmetical definition of area ("base X height") is
not needed and is never used; indeed, this
theory of application of areas, together
with the Book V theory of proportions, provides a completely adequate alternative to
the construction of the real numbers and
their use in plane rectilinear geometry. It merits considerable respect, and
gets it: the same (probably equally unreliable) story is found about Pythagoras sacrificing an ox to the discovery of a result on the application of areas
as is also told about the theorem on right angle triangles.
The golden section is constructed in Proposition 30:
To cut a given finite straight line in extreme and mean ratio,
and the method used there involves an elaboration of the theory called "application with excess." Fortunately, an easier construction is possible and
has already been given in Book II; and the manuscripts that we possess of the
Elements
contain a second, possibly interpolated, proof of VI, 30, referring
back to this earlier construction. Using this method, it is possible to bypass the use of proportion theory, and the elaborations of the theory of application of areas, and to give a direct definition and construction of the
golden section. This is now we shall proceed.
The Construction of the Line Divided in Extreme and Mean Ratio,
and Its Generalization
Book II, Proposition 11, describes how:
To cut a given straight line so that the rectangle contained by
the whole and one of the segments is equal to the square on the
remaining segment,
and we shall hereinafter adopt this as the definition of the extreme and mean
ratio. The construction is straightforward:
To construct the required point C on AB 9 draw the
square ABDE; take F to be the midpoint of AE, and G on
EA produced such that FG = FB. If ACHG is the square
with side AG9 then C cuts AB in mean and extreme ratio.
The verification of this is easy:
FG 2 = (AF + AG)
2
But
FG1
1
2
= AF + AC + 1AF • AC (Since AG = AC.)
AF2 + AB2. (By Pythagorasf theorem.)
FBZ
150
A GENERALIZATION OF THE GOLDEN SECTION
Therefore,
AC1
Subtract
then
IKE • AG = AE • AC from both sides,
AC1 = vl£ • CB = CS • B£>.
Q.E.F.
[May
+ 2AF • AG ?= AB1.
This proof can be read as if AF • AC 9 for example, represented the product
of two numbers, the lengths of AE and AC; or the purist can interpret AE • AC
as a rectangle with sides equal to the lines AE and AC and, using some obvious manipulations, check that the proof makes sense and is correct. This latter method is in the spirit of the techniques of application of areas, though
none of the subtle manipulations of that theory are needed.
It is clear that it must be the rectangle contained by the whole and the
lesser* segment that will be equal to the square on the greater
segment, since
the square on the lesser segment will fit inside the rectangle contained by
the whole and the greater segment and so it has smaller area. (The common notions at the beginning of Book I set out what are, in effect, the axioms of a
theory of equality and inequality of area or, more strictly, of content; and
Common Notion 5 states: The whole is greater than the part.)
We now describe the generalization that we call the nth order extreme and
mean patio, abbreviated to the noem ratio.
There is one such construction for
each integer n, and the golden section corresponds to the case n - 1; the implications of the construction are somewhat simpler for the case of even val^ues of n, and therefore we shall always illustrate the case of n = 3; and we
shall shortly introduce and use a consistent and general notation and terminology to describe the resulting configuration.
Start with the square ABDE on the given line
AB, and on AE produced as necessary, take points
F9
G9 H9-as
shown, with kAB = AE = EE = EG = GH9
etc.; then these points will be used in the construction of the 1st, 2nd, 3rd, 4th, etc. , extreme
and mean ratios. We always illustrate the case
of n- 3 and so, here, work from the point G. On
EA produced, take J such that GJ = GB; then the
square AJKCi defines the point C1 dividing AB in
the 3rd extreme and mean ratio.
-1st
2nd E
3rd G
4th H
1982]
A GENERALIZATION OF THE GOLDEN SECTION
151
The Definition and Properties of the Noem Ratio
We start with the basic defining property of the generalization, and show
that it is possessed by our constructed point.
Definition: The point C1 is said to divide AB in the noem ratio
(read: nthorder extreme and mean ratio) if, taking points C19 ..., Cn-i> Cn on AB such
that
ACX = 0^2
j
l
A
C1
_j
C„.x
I
Cn
l_
B
= ,• • • = Cn_1Cn , then Cn lies between A and B and AB • CnB = AC\.
Note that the latter condition implies that CnB is less than AC1; it will
be called the ''lesser segment" of the noem ratio. The greater segment of the
golden ratio generalizes two ways: to ACl9 which we call the initial segment
of the noem ratio; and to ACn, which we again call the greater segment of the
noem ratio. Care must be exercised in generalizing the results on the golden
section to make the appropriate choice. As remarked earlier, we shall always
illustrate the case of n = 3, and will always use the same letters to label
the points, calling the three division points C19 Cn-l9Cn9
so that their roles
will be clear. Proofs will be given for the general case, sometimes referring
to a phantom point C2 and adding a few dots "+ ••• + ."
Proposition:
noem ratio*
The point C19
described in the construction, divides AB in the
Proof: The figure illustrates the construction
for the case n = 3 . The proof is a straightforward
generalization of the proof given in the case n = 1,
and we can even use the same letters to identify the
vertices of the* figure.
As before,
FG2 = (AF
•
= AF2
2
But
FG = FB2
Therefore, AC\ + 1AF
But
2AF • AC1
(since AF = jAE9
+ AG)2
+ AC\ + 2AF • ACX.
= AF2 + AB2.
• Ad = AB2.
= nAE • ACX = AE • ACn .
and nAC1 =
ACn\
Hence
ACn < AB and, subtracting AE • ACn from
both sides, we see that AC\ = AB • CnB.
Q.E.F.
Book XIII of Euclid's Elements
contains the details of the construction of
the five regular "Platonic" solids, and a proof that these are the only regular solids; but it contains a lot more material besides that. In particular,
it starts with six propositions on the extreme and mean ratio, together with
alternative proofs of these results illustrating a method of "analysis and
synthesis." These propositions follow on in the style of Book II—-in particular, they do not explicitly need to use any more than the rudiments of the
152
A GENERALIZATION OF THE GOLDEN SECTION
[May
theory of application of areas—and they can easily be generalized to apply
to the noem ratios. We now alternate the enunciations of these Euclidean
propositions with their generalizations, interposing some general remarks.
(Later propositions of Book XIII describe relationships between the extreme
and mean ratio and pentagons, hexagons, decagons, icosahedra, and dodecahedra;
we shall not consider them here.)
XIII, Proposition 1. If a straight line be cut in extreme and
mean ratio, the square on the greater segment added to half of
the whole is five times the square on the half.
Paraphrase of Euclid's Proof:
If AB is cut in extreme and mean ratio at C,
and DA = h&B, then we prove CD2 = 5AD2.
Draw the squares on DC and AB, and complete
the figure as shown. (In addition to the Euclidean labelling of the vertices, we have also labelled the regions of the figure.)
We know that AB • CB = AC1 (Definition of mean
and extreme ratio)
i.e.,
AB 'AC = 2AD- AC (Since
AB
R — ZO
IS, i — *-> i "I" J 2 J
and
i.e.,
hence
Adding AD2
squares,
P + R = Q + S± +
JP,
"71
V
D*-
2AD)
S2.
and assembling the result into
DC2 = AB2 + AD2.
But AB2 = kAD2 , so DC2
5 AD2.
Q.E.D.
Remark: Our way, today, of considering the golden ratio is almost always to
identify it with the real number ^(/5+ 1); this and the following propositions
represent the closest approach we find in surviving Greek texts to this evaluation. For instance, this proposition implies that if AB = 2, then CD = /5
(i.e., the side of a square of area equal to the rectangle with sides AB and
5AB) so AC = /5 - 1, and the ratio is 2: ( A - 1) [= ^(/5 + 1):!].
Proposition 1': If a straight line be cut in the noem ratio, the square on the
initial segment added to n times half of the whole is n2 + 4 times the square
on the half.
Remark: It is standard Euclidean practice to handle such a general proof by
choosing a particular small value of n, typically n = 2, 3, or 4. The figures
for our proofs differ very slightly according as n is even or odd (a consequence of the occurrence of halves in the construction), with the case of even
1982]
A GENERALIZATION OF THE GOLDEN SECTION
153
n being slightly simpler. It is also standard Euclidean practice when there
are several different cases to a proposition only to consider the most complicated one. Therefore, our choice of n = 3 is in line with the Euclidean procedure. We shall9 however, use a general labelling system, writing C19 Cn_l9
Cn9 rather than C±, Cz> C3, and develop further the practice of labelling regions of the figure, using letters P, Q, R, etc., and suffixing to denote equal
regions, so P1 = P 2 = P 3 etc. Euclidean practice appears to be to label only
the vertices of the figure, working through the alphabet strictly in order of
occurrence in the setting-out and construction of the figure.
A final point in which our enunciation differs from Euclidean practice is
in referring to the chosen parameter n.
A more idiomatic expression, as rendered in English, might read:
Proposition 1": If a straight line be cut in the general extreme and mean ratio to some number, the square on the initial segment added to that number of
segments each equal to half of the whole is the square of that number increased
by four times the square on the half.
Purists might like to try a similar rephrasing of later generalizations!
Proof: If AB is cut in the noem ratio at C19 and DnA = -~AB9 then we prove
that DnCl = (n2 + 4)4P2.
Draw the squares on ~DnC\ and AB9 and complete the figure as shown:
S*
^1
•Vn
We know t h a t
i.e.,
and
i.e.,
D
AB • CnB
P
AB • AC1
R
n
_
=
=
=
=
x
D±
C
n-1
C
n
A
AC\
( D e f i n i t i o n of t h e noem r a t i o )
Q9
2AD1 • AC± ( S i n c e AB = Z4PX)
2S19
154
A GENERALIZATION OF THE GOLDEN SECTION
and hence,
P + nR = Q +
[May
2nS1.
Adding AD2 = n2T and assembling the result into squares, we get
DnC\ = AB2 + n2T
= (n2 + 4)T
(Since AB2 = 4T)
= (n2 + k)AD\.
Q.E.D.
The next propositions give the converses to these results. We start with
Euclid's enunciation:
XIII, Proposition 2. If the square on a straight line be five
times the square on a segment of it, then, when the double of
the said segment is cut in extreme and mean ratio, the greater
segment is the remaining part of the original straight line.
The Euclidean practice of never referring to a particular figure can make
the enunciations of propositions very cumbersome, and these propositions, together with the propositions of Book II contain some particularly awkward examples. In these cases, it is best to ignore the enunciation and proceed
directly into Euclid's proof of the proposition, where the setting-out will
give a more accessible explanation of the result. In this case we find, paraphrasing and adjusting the labelling to accord with our convention, that if
C and B are taken on a line DA produced with DC2 = 5DA2 and AB = WA9 then C
cuts AB in the extreme and mean ratio with AC the greater segment.
-L,
D
Proposition 2':
,
L
A
J
i
C
B
If a line DnA is divided equally into
and C\ and B are taken on DnA produced with
DnCl = (n2 + 4)Z?nD*_1
and
AB = ~pnA.=
2DnDn_19
then C-L cuts AB in the noem ratio with AC1 the initial segment.
J
Dn
L _
I
Z? n - 1 -
D1
I
A
I
I
Cx
Cn.x
I_ L
Cn
B
Proof: For both propositions we can construct the same figures as for the
preceding propositions and then read the previous arguments backwards. Q.E.D.
XIII, Proposition 3. If a straight line be cut in extreme and
mean ratio, the square on the lesser segment added to half of
the greater segment is five times the square on half of the
greater segment.
1982]
A GENERALIZATION OF THE GOLDEN SECTION
155
Proposition 3': If Cx cuts AB in the noem ratio, and ADn = ^AC1 (as shown),
then DnB2 = (n2 + b)AD\i i.e., the square on (the lesser segment CnB added to
half of the greater segment ACn) is equal to (n2 + 4) times the square on (half
of the initial segment AC±).
*1
Proof: Construct the figure shown,
where A9 B9 Cl9 Cn.l9 Cn are as usual,
s
and AD1 = D ^ ^
s
First observe that
DnCn = ACn
-ADn
- nACi - nADj,
= nAD1 ( S i n c e AC-L
24Z?!)
R2\
= ADn.
Hence,
DnB2 = Q + R1: + S
p
——hD,
C\
= Q + i?2 •••+ 5
1^_
{—
,
Cn
n-l
+ ACX (Since C1 divides AB in the noem ratio.)
= (n2, + 4)4£>2.
(Since g = £ n C 2 = n2AD{ and 4C X = 24Z?X.) Q.E.D.
XIII, Proposition 4. If a straight line be cut in extreme and
mean ratio, the square on the whole and the square on the lesser segment together are triple of the square on the greater
segment.
Proposition 4 1 :
Let AB be cut in the noem ratio at C± , then
AB2 + CnB2 = (n2 +
2)AC\9
i.e., the square on the whole and the
square on the lesser segment together
are (n2 + 2) times the square on the
initial segment.
R
#2
Proof: We have that CnB • AB = AC\,
i.e.,
Q1 + R~P.
Hence,
+ i?= 2P.
+ R +
Qi\
Adding AC2 = n 2 P to each side, and
assembling into squares, we see that
AB2 + CnB2 = (n2 + 2)ACl>
Q.E.D.
P
1
1
_J
156
A GENERALIZATION OF THE GOLDEN SECTION
[May
XIII, Proposition 5. If a straight line be cut in extreme and
mean ratio, and there be added to it a straight line equal to
the greater segment, the whole straight line has been cut in
mean and extreme ratio, and the original straight line is the
greater segment.
Proposition 5':
If C± cuts AB in the noem ratio, and An is taken on BA proproduced with DAn = ACi , then DB is cut in
the noem ratio by A.
duced with AnB
= nAB, and D on BAn
Note: In the Euclidean proposition, n = 1 and An = A; therefore, there is no
need to mention the first step of constructing the point An* After this step
the generalization states that, if there be added to AnB a line equal to the
initial segment, the whole BD has then been cut in the noem ratio, with the
line BAn being the greater segment, and so the original line BA the initial
segment.
Proof: Complete the figure as shown;
Now,
we want to show that DAn • DB = AB^
~TR
DAn • DA = P + Q1 +
and
Jn-i
+
«i
+
R
and, since C1 cuts AB in the noem ratio,
AB • CnB
i.e. ,
Hence
ACf,
P = R.
DAn • DB = AB2
D
A,
A
-A,
Cn
Ci
c
XIII, Proposition 6. If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational
straight line calleid apotome.
This result, together with its proof, generalizes directly to the noem
ratio, but an explanation of what it means depends on a knowledge of the long
and difficult Book X. It is perhaps worth noting that Euclid uses the words
"rational" and "irrational" here in completely different sense from our modern
usage: a short, though oversimplified explanation is that when a unit line p
has been chosen, then anything of the form /jK-• p (where p and q are integers)
is called rational; anything not of that form is an irrational; and an apotome
is an irrational line that can be expressed as a difference of two rational
lines,
p-/l
1982]
A GENERALIZATION OF THE GOLDEN SECTION
157
Ratio in Eudlid's Elements
It is a curious and remarkable fact that ratio is not defined either in
Eudlid's Elements,
or anywhere else in the surviving corpus of Greek mathematics. All that we have is a vague description of the word at Book V, Definition 3:
A ratio is a sort of relation in respect of size
between two magnitudes of the same kind.
What is defined (at Book V, Definition 5) is proportion, which is a relation
that may or may not hold among four magnitudes, aibiioid;
and we can think of
it, and appear to be encouraged in this by Euclid, in terms of the equality of
two "ratios." An examination of the scanty surviving evidence of pre-Eudlidean
mathematics, and a reinterpretation of some of the books of the Elements
has
led me to suggest that ratio might have been defined, in the period before the
development of the abstract proportion theory that we find in Book V of the
Elements,
by a process based on the "Euclidean" subtraction algorithm. (Actually, what little evidence we have indicates that the person who realized the
importance of the procedure might have been Theaetetus, a colleague and friend
of Plato, so the "Theaetetan subtraction algorism" might be a more appropriate
name; here, I have also corrected what the OED calls a "pseudo-etymological
perversion. . . in which algorithm is learnedly confused with Greek apiOyos.")
Let me illustrate this by describing the operation of the procedure on two
lines a 0 and a1.
Suppose that a1 goes into aQ some number n 0 of times, leaving a remainder a2 less than a1; and then a2 goes into a± some number nx of
times, leaving a remainder a 3 ; etc. Then the ratio aQia
will be defined
by
the sequence of integers [n0, n19 n2,
...].
If, at any stage, a remainder is zero, the process terminates, and this is
characteristic of commensurable ratios. Among incommensurable ratios, with
nonterminating expansions, the simplest will be the ratio in which, at each
step, the smaller magnitude goes once into the larger magnitude, leaving a remainder for the next step, thus giving the ratio [1, 1, 1, . . . ] . This is the
golden ratio, as can immediately be deduced from the figure of the regular pentagon of which the diagonals, which form an inscribed pentagon, cut each other
in the golden ratio (this is explicitly proved at XIII, 8, but the result is
implicit in the construction of the pentagon given at IV, 11); or it can easily be deduced from the defining property of the ratio. What we have been
constructing here are the next simplest incommensurable ratios, of the form
[n, n, n, . . . ] , in which, at each stage, the smaller magnitude goes n times
with a remainder into the larger magnitude. By using a bit of algebra we can
easily work out the numerical value 9 of this ratio, since
= n+
e = n + FTT
e'
n + ...
so 9
2
- n6 - 1 = 0, and, taking the p o s i t i v e r o o t ,
6 = kW(n2
+ 4) + n).
[Alternatively, we can read off from the construction that
6 = AB/AC1 = 2/(/(n2 4- 4) - n) = hW(n2
+ 4 ) + n).]
158
A GENERALIZATION OF THE GOLDEN SECTION
[May 1982]
This explains the occurrence of the number 5, generalizing to n 2 + 4, the
halves, and the addition and subtraction of segments in the propositions that
we have been proving.
It is possible to extend the construction, and thus describe a procedure
for constructing any ratio that eventually becomes periodic, though the longer
the period, the more involved becomes a perliminary calculation of two parameters needed in the construction. (One of these parameters describes the location of the initial point on the left-hand edge of the square on AB, in our
diagram; the other describes the position of an auxiliary point Br on AB; the
construction then continues from these two points as before.) Further details
of these constructions, together with details of the historical and mathematical ideas that fill out, explain, and set in context these remarks, are given
in the papers [7], [8], and [9].
I do not know whether any of the noem ratios, with n >_ 3, occur in any
regular or semiregular figure, generalizing the appearance of the golden section in the pentagon and other figures, and the ratio [1, 2, 2, 2, ...] of the
diagonal and side of a square.
References
1. I. Adler. "A Model of Contact Pressure in Phyllotaxis." Journal of Theoretical
Biology 45 (1974):1-79.
2. E. T. Bell. "The Golden and Platinum Proportions." National
Mathematics
Magazine 19 (1944):21-26.
3. T. Brunes. The Secrets
of Ancient
Geometry—And
Its Use. 2 vols. Copenhagen: Rhodos, 1967.
4. H. S. M. Coxeter. Introduction
to Geometry.
2nd ed. New York: Wiley,
1969.
5. H. S. M. Coxeter. "The Role of Intermediate Convergents in Tait?s Explanation of Phyllotaxis." Journal of Algebra 20 (1972):168-175.
6. Euclid. Elements.
Ed. and Trans, by T. L. Heath. 3 vols. 2nd ed. Cambridge University Press, 1926; repr. New York: Dover, 1956.
7. D. H. Fowler. "Ratio in Early Greek Mathematics." Bulletin
of the American Mathematical
Society,
New Series 1 (1979):807-846.
8. D. H. Fowler. "Book II of Euclid*s Elements,
and a Pre-Eudoxan Theory of
Ratio." Archive
for History
of Exact Sciences
9. D. H. Fowler. "Book II of Euclid!s Elements,
Ratio, Part 2: Sides and Diameters." Archive
ces,
to appear.
10. H. L. F. von Helmholtz. On the
11.
12.
13.
14.
Sensations
22 (1980):5-36.
and a Pre-Eudoxan Theory of
for History
of Exact
Scien-
of
Tone as a
Physiological
Foundation
for the Theory of Music.
First German edition (1986); English
translation by A.J. Ellis of the fourth German edition (1877), with numerous additional notes and appendices, 1885; repr. New York: Dover, 1954.
G. Sarton. "Query No. 130: When did the Term fGolden Section* or Its
Equivalent in Other Languages Originate?" Isis
42 (1951):47.
DfA. W. Thompson. On Growth and Form. First edition 1917; Second edition
1942. Cambridge University Press.
R. Wittkower. "The Changing Concept of Proportion." Daedalus (Winter,
1960):199~215.
L. Churchin & R. Fischler. "Hero of Alexandrians Numerical Treatment of
Division in Extreme and Mean Ratio and Its Implications." Fhoenix 35
(1981):129-133.
*****
THE FIBONACCI SEQUENCE IN SUCCESSIVE PARTITIONS
OF A GOLDEN TRIANGLE
ROBERT SCHOEN
University
of Illinois
at Urbana-Champaign, Urbana, IL 61801
(Submitted February 1981)
A Golden Triangle is a triangle with two of its sides in the ratio (J>:1,
w h e r e <f> is the Fibonacci R a t i o , i.e., (J> = ^(1 + / J ) ^ 1.618. L e t AABC b e a
triangle w h o s e sides are a,b,
and o and let a/b = k > 1. Bicknell and Hoggatt
[1] have shown that (1) a triangle w i t h a side equal to b can b e removed from
AABC to leave a triangle similar to AABC if and only if k = cj), and (2) a triangle similar to AABC can b e removed from AABC to leave a triangle such that
the ratios of the areas of AABC and the triangle remaining is k if and only if
& = <(>.
Unlike the Golden Rectangle whose adjacent sides are in the ratio cj>:l (or
l:cj)), the Golden Triangle does not have a single shape. T h e diagonal of a G o l den Rectangle divides it into two Golden Triangles whose sides are in the r a tio 11cj):/cj)2 + 1. The most celebrated Golden T r i a n g l e , w h i c h can b e found in
the regular pentagon and regular decagon, has angles of 3 6 ° , 7 2 ° , and 72° and
sides in the ratio 1 K p : ^ .
In general, Bicknell and Hoggatt demonstrated that
a Golden Triangle can b e constructed w i t h sides in the ratio l:cj):c7, w h e r e (J)"1
< G < cj)2. Figure 1, adapted from their presentation, shows Golden Triangle
CGH.
Line GH is constructed to b e of length r$ (r > 0) and line CG to b e of
length rcj)2. L i n e CG is twice divided in the Golden Section by points E and D,
w i t h CE = DG - v and ED = r/§.
A Golden Triangle is formed whenever H is a
point on the circle whose center is G and w h o s e radius is EG. L i n e DH produces
ADGH ~~ ACGH, and ACDH w h o s e area is 1/cf) times the area of ACGH. In general,
ACDH is not similar to ACGH.
N o n e t h e l e s s , ACDH is also a Golden Triangle, as
CH/DH = <f> [ 1 ] .
The present paper will explore the consequences of successively partitioning Golden Triangles. To b e g i n , let us show that ACDH can b e partitioned into
two triangles, one similar to itself and the other having an area 1/cf) times
its own area. If line DJ is drawn parallel to line GH, one can readily v e r i fy that ADHJ is similar to ACDH.
(Alternatively, w e could h a v e chosen point
J so that CH/CJ = cj). Lines DJ and GH would then b e parallel, because CH/CJ =
CG/CD.)
W e n o w need to show that the ratio of the area of ACDH to the area of
ACDJ is cj). If w e designate the area of ACGH by S, the area of ACDH is
S/$
[ 1 ] . Since DJ is parallel to GH, ACDJ ~ ACGH.
The ratio CG/CD = <|>, hence the
area of ACDJ is S/$2.
Accordingly, the ratio of ACDH to ACDJ is S/<$> divided
by S/$2,
or cj). Since 5/cJ) - 5/cj)2 = S/$3 , w e find that the area of ADHJ is S/cJ)3.
W e can note several other relationships. Two additional Golden T r i a n g l e s ,
ACDJ and ADHJ, are produced so that ACGH is partitioned into three mutually
exclusive Golden Triangles. M o r e o v e r , ACDJ is congruent to ADGH.
They are
similar, as both are similar to ACGH and both have areas equal to 5/cJ)2.
Moving beyond the Bicknell-Hoggatt demonstration and its immediate i m p l i c a t i o n s , w e can show h o w successive partitions of Golden Triangles generate
Fibonacci sequences. L e t us repeat the above partitioning, subdividing all of
the larger triangles produced in the previous partition. The partitions can
be carried out in a manner analogous to the w a y in which ACDH w a s partitioned.
159
THE FIBONACCI SEQUENCE IN SUCCESSIVE PARTITIONS
OF A GOLDEN TRIANGLE
160
^
GH
DH
DG
DJ
[May
HJ
a = tDCH = tDHG = L.KDJ
hCDJ = LDGH
FIGURE 2. The General
Golden
Triangle
For example, ACDJ can be split into two Golden Triangles by a line through
point J parallel to ED. The resultant line is JE, which has a length equal to
DH/$ and divides line CD in the Golden Section. As we proceed, the number of
triangles and their areas are as follows:
Partition
Number
(n)
Fibonacci
Number
(F )
1
2
2
1
3
3
2
5
4
5
Area of Triangles
1
0
1
Number
of
Triangles
3
5
8
13
S/<f>,
Sl^1
2
S/cf) , S/cf)2, S/cf)3
S/<$>\ S/cj)3, S/$\
S/^
(5 t r i a n g l e s ) , S/($> (3 t r i a n g l e s )
5
(8 t r i a n g l e s ) , S/$6
S/$
S/§
Sl§n
S/cj)\
k
5
(Fn+l
t r i a n g l e s ) , S/$n+1
(5 t r i a n g l e s )
(Fn t r i a n g l e s )
1982]
THE FIBONACCI SEQUENCE IN SUCCESSIVE PARTITIONS
OF A GOLDEN TRIANGLE
161
The total number of triangles and the number of larger and smaller triangles
increase in Fibonacci sequence. Figure 2 shows the five triangles produced by
the third partition. The eight triangles produced by the fourth partition result from subdividing the three larger triangles in the figure. Because partitioning produced triangles whose areas, relative to the area of the partitioned triangle, are l/(j> and l/(j)2, the pattern is perpetuated.
S = Area kCGH
ACEJ ~ kDHJ ~ kDHK
bDEJ = kDGK
FIGURE 2.
The Partition
of a Golden Triangle
Five Golden
Triangles
into
By a repetition of the earlier demonstrations, it can be seen that every
triangle that results from the partitioning is similar to one of the two Golden Triangles produced by the first partition (i.e., the partition effected by
line DE), and that all triangles of the same area are congruent. Every triangle has an area equal to S/fy'1, for some integer i. For triangles similar to
ACGH9 i, is even, while for triangles similar to &CDH9 i is odd. Corresponding sides of triangles with areas S/<^i and 5/(J^ + 2 are in the ratio <\>:1. The
total area of the larger triangles relative to the total area of the smaller
is §Fn+1/Fn
after the nth partition, and that ratio approaches (j)2 as n becomes
large.
Each partition illustrates the equation for powers of (J), i.e.,
(1)
Dividing (1) through by cf)n, we have
V*-Fn$+Fn_x.
THE FIBONACCI SEQUENCE IN SUCCESSIVE PARTITIONS
OF A GOLDEN TRIANGLE
162
[May
which expresses the area of a unit Golden Triangle as the sum of the areas of
partitioned triangles. For example, with n = 4, we have the situation before
the fourth partition, shown, in Figure 2, where
a - 3S , 2S
(3)
6
Multiplying (3) by $** IS
" IF+ TV*
gives
3<|> + 2.
(4)
Let us move from the general case to the special case where the two Golden
Triangles formed by the first partition are similar to &CGE, and hence to one
another. In that case, the triangles must be "Fibonacci Right Triangles," with
sides in the ratio (J):c()3/2:(j)2. To demonstrate that, consider triangles CDE and
DGE in Figure 1. From (1) we know LDCH = LDEG.
If triangles CDH and CGH are
similar, LCED must equal L.DGE because i-CED + LCEG.
Since ACDE ~ M)GE and we
have established equalities between two of their three angles, we must have
LCDH = LGDH.
A S LCDE and LGDE sum to 180° , both of those angles equal 90° and
line BE is an altitude. With LCEG = LCED + L.DEG and LDEG = LDCE,
we have LCEG =
L.CED + tDCE.
In right triangle CDE, L.CED and L.DCE sum to 90° , hence i^CEG must
be 90° . As Figure 1 was constructed with GE = rcj> and CG - r(j)2, applying the
Pythagorean Theorem yields r2$h
= v2§2
+ CE2, and thus we find CE = P(J)3/2.
The Fibonacci Right Triangle has been examined by a number of writers.
Ghyka [2] identified it as one of the three most significant nonequilateral
triangles. He noted that it was sometimes called the "Great Pyramid" triangle
because its proportions are found in the Great Pyramid of Cheops, or the triangle of Price, after W. A. Price, who proved that it is the only right triangle whose sides are in geometric progression (i.e., if the sides of a triangle
are 1, k9 and k2, k = /cjT is the only positive real solution that satisfies the
Pythagorean equation 1 + k2 = kh).
Hoggatt [3] noted that the altitude of a
Fibonacci Right Triangle produced two Fibonacci Right Triangles that were
"five parts congruent," that is, were similar and had two (but not three) sides
of equal length. The Fibonacci Right Triangle is related to mean values, in
that the harmonic, geometric, and arithmetic means of two positive numbers form
a right triangle (the Fibonacci Right Triangle) if and only if those numbers
are in the ratio (J)3:l [4]. In successive partitions of Fibonacci Right Triangles, all line segments are in Fibonacci proportions, as they are all multiples of (j)i/2, with i an integer [5]. A multiply partitioned Fibonacci Right
Triangle thus presents a striking geometric pattern. An example is given in
Figure 3, which shows the 13 Fibonacci Right Triangles that result from five
partitions of the original triangle.
FIGURE 3.
Five Partitions
of a Fibonacci
Right
Triangle
1982]
THE FIBONACCI SEQUENCE IN SUCCESSIVE PARTITIONS
OF A GOLDEN TRIANGLE
163
In summary, successive partitions of a Golden Triangle provide a multifaceted geometric representation of the Fibonacci sequence. The triangles
described above are Fibonacci in three different ways because they are in Fibonacci proportions with regard to their numberss their areas, and the lengths
of their sides. Golden Triangles not only embody the Fibonacci ratio, they
also carry within them the ability to generate Fibonacci sequences.
References
1.
2.
3.
4.
5.
Marjorie Bicknell & Verner E. Hoggatt, Jr. "Golden Triangles, Rectangles,
and Cuboids." The Fibonacci
Quarterly
7, no. 1 (1969):73-91.
Matila Ghyka. The Geometry of Art and Life.
New York: Dover, 1977 (brig.
ed.5 1946).
Verner E. Hoggatt, Jr.. Fibonacci
and Lucas Numbers.
Boston: Houghton Mifflin, 1969.
Robert Schoen. "Means, Circles, Right Triangles, and the Fibonacci Ratio."
The Fibonacci
Quarterly
19, no. 2 (1981):160-162.
Joseph L. Ercolano. "A Geometric Treatment of Some of the Algebraic Properties of the Golden Section." The Fibonacci
Quarterly
11, no. 2 (1973):
204-208.
kkkkk
GEOMETRY OF A GENERALIZED SIMSON'S FORMULA
University
A. F. HORADAM
of New England, Armidale,
N.S.W.
(Submitted
April
1981)
1.
2351,
Australia
Introduction
Articles of a geometrical nature relating to recurrence sequences have
appeared in recent years in this journal (e.g. [1], [2], [6]).
The purpose of the present paper is to consider the loci in the Euclidean
plane satisfied by points whose Cartesian coordinates are pairs of successive
numbers in recurrence sequences of a certain type. Readers might plot some
points on the resulting noncontinuous curves (conies).
Extension to higher-dimensional space is briefly discussed.
2.
The General Conic
Begin by defining [4] the general term of the sequence {wn(a> b; p, q)}
(1)
wn+2
= pwn+1
- qwn,
as
wQ = a, w1 = b,
where a, b, p, and q belong to some number system, but are usually thought of
as integers. Write [4]
e = p a b - qa2 -
(2)
b2.
Now [4]
W W
(3)
n n+2
-
W
n +1 =
^*>
?
which is a generalization of Simson s formula
(4)
p p
- F2
(-nn+1
occurring in the Fibonacci sequence {Fn} = {wn(0,
1; 1, -1)}.
Equation (3) generalizes the famous geometrical paradox associated with
(4). For the details, see [5].
From (1) and (3), we obtain
(5)
qw1 + w2
Next, put Wn = x9 Wn+1 = y.
(6)
- pw w
+ eqn
= 0.
Then, by (5),
qx2 + y2 - pxy + eqn
= 0.
This equation represents a conic in rectangular Cartesian coordinates
Or, y). Anticlockwise rotation of axes through an angle
164
[May 1982]
GEOMETRY OF A GENERALIZED SIMSON'S FORMULA
165
eliminates the xy term and produces the canonical form of the conic (an ellipse if p 2 < kq9 a hyperbola if p 2 > kq> where the degenerate cases are excluded) . Equation (6) is also obtainable by laborious reduction of the general
equation of a conic using the uniqueness of a conic through 5 given points.
3.
!•
Some P a r t i c u l a r Cases
<7 < 0 (Hyperbolas)
(a) p = 19 q = -1: Substituting in (6) yields the two systems (n even,
n odd) of rectangular hyperbolas
x2
(7)
- y2 + xy = e1(-l)n
(e1
= a2 - b2 + ab),
asymptotes of which are the perpendicular lines
(8)
y = ax,
in which a =
y = —ar,
, the positive root of t2
- t - 1 = 0.
For the
Fibonacci
sequence
(a = 0, b = 1) and the Lucas sequence
(a = 2, 2? = 1), it follows that
ex = -1 and 5, respectively. These Fibonacci-type
curves (7) approach their
asymptotes remarkably quickly.
With a fixed eY in (7), a hyperbola for which n is odd (even) may be transformed into the corresponding hyperbola for which n is even (odd), by a reflection in y = x followed by a reflection in the z/-axis (x-axis).
(b) p = 2, q = -1: For the Pell
sequence
(a = 0, b = 1), (6) gives
x»2 - y2 + 2*2/ = (-l)n+ 1 ,
(9)
rectangular hyperbolas with perpendicular asymptotes y = kx, y = -j-x,
where
fc = 1 + /2" is the positive root of t2 - 2t - 1 = 0.
Gradients of the perpendicular asymptotes of the hyperbolas (6) for which
p > 0, q = -1 are given by the roots of t2 - pt -•1 = 0.
II.
4 > 0
Equation (6) now represents ellipses if kq > p2 and hyperbolas if hq < p .
For example, the loci for the Fermat
sequences
{wn(0,
1; 3, 2)}
and
|u„(|,.2; 3, 2 U
are hyperbolas (one point for each n)
2x2 + y2 - 3xy = 2n
(10)
and
(11)
2x2 + y2 - 3xy = - 2 n _ 1 .
Further, for the Chebyshev
{wn(l,
sequences
2X; 2A5 1)}
and
{wn(2,
2X; 2\,
1)},
GEOMETRY OF A GENERALIZED SIMSON'S FORMULA
166
[May
where A = cos 9, we obtain the ellipses
x2 + y2 - 2\xy
(12)
and
(13)
III.
x2 + y2 - 2\xy
=1
= 4 - 4A2.
Degenerate Case
When X = 1 in (12), i.e., for the sequence {w (1, 2; 2, 1)}, of
we have the degenerate curve x2 + y2 - 2xy = 1, i.e., the line
integers,
x - y = -1.
No values of x + y,
pairs of odd integers
as defined, satisfy the equation x - y = 1.
and of even integers,
generated by
3; 2, 1)}
{wn(ls
and
Successive
4; 2, 1)},
{wn(2,
respectively, satisfy the line
(15)
x - y = -2.
4.
Extension to Higher Space
Equations of the third, fourth, and higher degrees that are based on second-order recurrences like (1) (see, e.g. [3], [4]) cannot yield any nondegenerate loci in spaces of dimension greater than two.
For three-dimensional (nonprojective) space, it is necessary to consider
third-order recurrence relations, of which the simplest is
(16)
P n+ 3 = Pn-+2 + P n + 1 + P „
in
> 0).
Waddill and Sacks [8] have established the following relation for {Pn} corresponding to the Simson formula (4) for {Fn}:
p2
p
n+3 n
+
p3
+
n+2
p2
p
_ p
n + 1 n+h
p
n+h
P • - 2P ' P
n+2
n
n+3
P
n+2
n+1
(17)
= Pi + 2P* +P\
+ 2P\P1 + 2PQPl + P2QP2 - 2P1P\
- 2P 0 P X P 2 - PQP22.
Putting P 0 = 0, ?! = P 2 = 1 and P 0 = 1, Px = 0, P 2 = 1 they obtained their
sequences {Kn} and {Qn}$ respectively:
(18)
{Kn}i
0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, ...;
(19)
{Qn}:
1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, ... .
Letting Pn=x,
manipulation,
(20)
Pn+1=sy>
^n + 2 =
z
in
x3 + 2y3 + s 3 + 2x2# + 2xy2
(17)»
- 2yz2
we
derive, after some algebraic
+ x2z
- xz2
- 2xz/s = A,
where A = 1 for {#„} and .4 = 2 for {$ n }. Equations (20) represent cubic surfaces in Euclidean space of three dimensions.
1982]
GEOMETRY OF A GENERALIZED SIMSON'S FORMULA
167
More general forms of (16) would lead to extremely cumbersome equations.
Observe that if we label any three successive numbers in (18) as x, y9 z,
and the corresponding three numbers in (19) as X, Y9 Z, then we perceive that
X = y - x9 Y - z - y9 Z = x + y.
Fourth- and higher-order recurrences should produce equations corresponding to (17) which are generalizations of Simsonfs formula (4). While (17) is
not a pretty sight, the mind boggles at the prospect of further extensions,
which we accordingly do not investigate. But the general pattern seems clear:
a recurrence of the nth order ought to lead to a hypersurface (of dimension
n - 1) in Euclidean n-space.
5.
Concluding Comments
a. For the sequence {wn(l9
a; 1, -1)}, e = 0 [see (1), (2), (7)] and the
curve (7) degenerates to the line-pair x2 + xy - y2 = 0.
b. Graphing the Fibonacci numbers Fn against n reveals that they asymptotically approach the exponential values
C. M. H. Eggar, in "Applications of Fibonacci Numbers" [The
Mathematical
Gazette
63 (1979):36-39], refers to (7), in the case where e1 = 1, though his
context is nongeometrical.
d. Interest in the theme of this article was stimulated by a private communication to the author in 1980 by L. G. Wilson, who determined the vertex of
the hyperbola (7) for the Fibonacci sequence, but only in the case where n is
odd, namely, x = 0..92.0442065...., y = 0.217286896... . He also calculated the
angle of inclination of the axis of this hyperbola to the x-axis, namely,
13.28252259... degrees [ (= 13° 17f) = tan^/B" - 2)].
Furthermore, Wilson briefly investigated the geometry of the third-order
sequence {Tn}i
(21)
0, 2, 3, 6, 10, 20, 35, 66, ...,
defined in Neumann-Wilson [7] by
(22)
Tn+3 = T n + 2 + Tn + i +Tn
+ (-l)n
(n >_ 0 ) .
References
1.
2.
3.
4.
5.
W. Gerdes. "Generalized Tribonacci Numbers and Their Convergent Sequences."
The Fibonacci
Quarterly
16, no. 3 (1978):269-275.
D. Hensley. "Fibonacci Tiling and Hyperbolas." The Fibonacci
Quarterly
16, no. 1 (1978):37-40.
V. E. Hoggatt. "Fibonacci Numbers and Generalized Binomial Coefficients."
The Fibonacci
Quarterly
15, no. 4 (1967):383-400.
A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci
Quarterly
3, no. 3 (1965):161-176.
A. F. Horadam. "Special Properties of the Sequence Wn(a9 b; p9 q ) . "
The
Fibonacci
Quarterly
5, no, 5 (1967):424-434.
168
6.
7.
8.
AN ENTROPY VIEW OF FIBONACCI TREES
[May
D. V. Jaiswal. "Some Geometrical Properties of the Generalized Fibonacci
Sequence." The Fibonacci
Quarterly
12, no. 1 (1974):67-70.
B. H. Neumann & L. G. Wilson. "Some Sequences Like Fibonacci's." The Fibonacci Quarterly
17, no. 1 (1979):80-83.
M. E. Waddill & L. Sacks. "Another Generalized Fibonacci Sequence." The
Fibonacci
Quarterly
5, no. 3 (1967):209-222.
*****
AN ENTROPY VIEW OF FIBONACCI TREES*
Shizuoka
YASUICHI HORIBE
University,
Hamamatsu, 432,
(Submitted
May 1981)
Japan
Abstract
In a binary tree with n terminal nodes weighted by probabilities pl9 .. .,
Pn9 ^Pi = 1 > it: ^ s assumed that each left branch has cost 1 and each right
branch has cost 2. The cost a^ of terminal node pi is defined to be the sum
of costs of branches that form the path from the root to this node. The sum
£Ptai *-s c^lled the average cost of the tree. As a top-down tree-building
rule we consider ^-weight-balancing
which constructs a binary tree by successive dichotomies of the ordered set pi, ..., pn according to a certain weight
ratio closely approximating the golden ratio. Let H = H(pl9
..., pn) = -Zp^
log pi be the Shannon entropy of these probabilities. The ijj-weight-balancing
rule is motivated by the fact that the entropy per unit of cost
H(x9
for the division x : (1 - x)
1 - x)/{l
• x + 2 • (1 - x))
of the unit interval is maximized when
x = i|> = (/5 - l)/2,
the golden cut point. It is then shown that the average cost of the tree built
by ijj-weight-balancing is bounded above by #/(-log ip) + 1, if the terminal nodes
have probabilities p1, ..., pn , p x _> ••• > pn, from left to right in this order in the tree. If Pj + 1/Pj ^ (l/2)ip for each j , the above bound can be improved to El {-log \p) + ijj. For the case p x = ••• = p n , we obtain the following
results. The ^-weight-balancing constructs an optimal tree in the sense of
minimum average cost and constructs the Fibonacci tree of order k when n = Fk9
the kth Fibonacci number. The average cost of the optimal tree is given exactly. Furthermore, for an arbitrarily given number of terminal nodes, the
ip-weight-balanced tree is also "balanced" in the sense of Adelson-Velskii and
Landis, and is the highest of all balanced trees.
We will discuss some properties of Fibonacci (Fibonaccian) trees in view
of their construction by an entropic weight-balancing, beginning with the following preparatory section:
*This paper was presented
at the International
Theory, Budapest,
Hungary, August 24-28,
1981.
Colloquium
on
Information
1982]
AN ENTROPY VIEW OF FIBONACCI TREES
1.
169
Binary Tree with Branch Cost
Let us consider a binary tree (rooted and ordered) with n - 1 internal
nodes (branch nodes) and n terminal nodes (leaves) [6]. An internal node has
two sons, while a terminal node has no sons. A node is at level £ if the path
from the root to this node has £ branches. The terminal nodes are assumed to
be associated, from left to right, with probabilities or weights p 2 , ..., pn,
Y,pi = 1. We assume, furthermore, that every left branch has unit cost 1, and
every right branch has cost o (>. 1). A node is then associated with two numbers, probability and cost; the probability of an internal node is defined to
be the sum of probabilities of its descendant terminal nodes, and the cost of
a node is defined to be the sum of costs of branches that form the path from
the root to this node. The root, then, has probability 1 and cost 0. Sometimes, for simplicity, a node will be named by the associated probability. We
define the average cost of a tree as
n
c = E ViZi*
i =l
where a± is the cost of the terminal node pi . Since we interpret C as the
average cost required to get to a terminal node by tracing the corresponding
path from the root, C measures a global goodness of the tree: for fixed n, c9
p x , . . . , pn , the smaller C is, the more economical the tree is. If we view
the binary tree, for example, as representing a binary code consisting of n
codewords with code symbols 0 of duration 1 (corresponding to the left branch)
and 1 of duration e (to the right) for the given source alphabet having letterprobabilities p19 ..., p , then C is the average time needed to send one source
letter.
An internal node will be called internal node J, l<_j<_n - 1, if its left
subtree has p- as the rightmost terminal node> (The leftmost terminal node of
its right subtree is then Pj+1>) Let us denote by Lj and Rj the probabilities
of the left and the right sons of the internal node j , respectively. Put
T
=
j
L
J
+
R
J
»
which is, of course, the probability of the internal node j .
We give here three general relations—(1), (2), and (3)—for use in later
sections. First we have
n-l.
(1)
C = £ (Lj + cRd).
J=I
This is seen by observing that the cost 1 [resp. a] of the left [right] branch
that connects the internal node j and its left [right] son contributes 1 • Lj
[c ' Rj] to C.
Second, let
n
H E ff(Pl,-..., pn) = -Y.Vi
log Vi
i =l
be the Shannon entropy.
(2)
(Logarithms will always be to the base 2.)
"-1
(L,
Rj\
E = E TjH[^,
-£).
We have
170
AN ENTROPY VIEW OF FIBONACCI TREES
[May
This is the well-known binary branching property of the entropy [4, 9 ] , The
entropy is known as a very appropriate function to measure the uncertainty,
the uniformness, or the randomness, of the probability distribution pl9 ...,
pn.
It is this aspect and the branching property that relates the entropy to
the economical structures of the trees having weighted terminal nodes, as will
be seen in the following sections.
Third, letting bj be the cost of the internal node j , we have
Lemma 1 :
f x = E&j + <n - DC1 + *>•
o)
Proof (by induction on n ) : When n = 1, (3) is trivially true. Consider
an arbitrary tree with n + 1 terminal nodes. At the maximum level there exist
two terminal nodes that are sons of the same internal node, say A:. Merge these
nodes into k to obtain a tree having n terminal nodes. The decrease in the
total cost of terminal nodes due to this merging is given by
(bk + 1) + (bk + o) - bk = bk + (1 + a).
On the other hand, the decrease in the total cost of internal nodes is
This completes the proof.
2.
bk.
Weight-Balancing and "Discrete" Golden Cut
A binary tree can be viewed as a pattern of successive choices between the
left and the right branches started from the root in order to look for a terminal node. The uncertainty per unit of cost, removed by the choice at the
internal node j , is measured by
H(EL
?L\
'•£)•••&)•
So the tree that maximizes this quantity at each step can be expected to have
a small average cost C. Of course, the successive local optimizations of this
type will not necessarily lead to a global minimization of the average cost.
Nevertheless, we will be concerned with this process because it is interesting
in its own right.
In the case o = 1, the above quantity reduces to H(Lj /Tj , Bj/Tj) , which
becomes maximum when \Lj - Rj\ is minimum, i.e., when
Lj - p. 12 < Tj/2
<LJ:+
Pj +
1/2,
for fixed Tj . The rule for constructing a tree in a top-down, level-by-level
manner, such that at each step Lj and Rj are made as equal as possible, is
called "weight-balancing." The binary code corresponding to the tree thus
built by weight-balancing under the monotonocity condition pi J> • • •__> pn is
1982]
AN ENTROPY VIEW OF FIBONACCI TREES
171
known as the Shannon-Fano code ([3], [9], see also [11]). This code is not
necessarily optimal (in the sense of minimum average cost), but it is almost
optimal, and satisfies
C <H + (1 - 2pn)
(see [4]). Henceforth,
we assume p, _> • • • _> p .
In order to generalize the above weight-balancing rule to the general c9
we naturally maximize the function
Hx
'
) - X \ ,
x + c(l - x)
(4)
0<x<
1.
— —
Let X be the maximizing value of #. By differentiating, X is the unique positive root of x° - 1 - x.
The maximum value of the function is -log X. Considering X = X(c) as a function of c9 we have X(l) = 1/2, X(e) is strictly
monotone increasing, and X(c) •> 1 as c ->• °°. Now define X-weight-balancing
as
a rule for constructing a tree satisfying
(5)
Lj - (1 - \)p.
< XTd <Lj
+Xp.+1
for each internal node J = 1, . .., n - 1. Recently, K. Mehlhorn has taken up
a similar rule to study search trees [8]. We shall be confined
especially
to
the case c = 2, where the "X-cut" X : (1 - X) of the unit interval becomes the
golden cut, since we have X(2) = (/5* - l)/2 = 0.618... . We denote this number by IJJ, its inverse ty ~1 = cj) being commonly called the golden ratio, and
ty2 = 1 - ty, -log ip = 0.694... . [Conversely, if x = ty maximizes (4), then c
must be 2.]
3.
Bounds on the Average Cost
For a reason that will be clear in the next section, trees constructed by
ty-weight-balancing may be called "Fibonaccian trees.11 In this section we find
entropic bounds on the average cost of Fibonaccian trees. Since we are treating c- 2, and -log \\) is the maximum value of H(x, 1 - x) / (2 - x) , the function
f(x)
= (-log i|0(2 - x)
1 - x)9
- H(x9
0<x<
1,
is nonnegative.
Theorem 1: • . H , < C < •., H , + (1 - p n ) . [Note that #/(-log ty) is the en—
-log ip —
- -log Ip
^n
tropy with respect to the log-base (f), i.e., #/(-log ip) = - Ep^ log f p^ . ]
Proof: The proof technique is that used in [5]. Consider the difference
(-log ty)C - H. From (1) and (2) in Section 1, we have
n-l
i-iognc
where
- H = £ ^-,
J = l
w X i * «(£•*•£)-<£.£)-*/(£)
172
AN ENTROPY VIEW OF FIBONACCI TREES
[May
The fact f(x) ^ 0 implies the left-side inequality to b e proved. (This lower
bound is well known, see [ 2 ] , and is valid for any tree, as w e see from the
proof.) To prove the upper bound, split (5) with X - \p into two cases, for
each j :
Case 1:
(6)
Lj - (1 - iJOp/ < Wd
<Lj.
Equation (6) leads us to ip <.Lj/Tj
< 1. The function f(x)
downward, and /(i|0 = 0, f(l)
- -log I|J. Hence,
f i x ) 1 (-log Wf-E-jjJ
is clearly convex
±f^P<x<l.
Therefore,
^
= ^ j ^ - j < (-log if;)-J
1 - ij,
But by the left-side inequality of ( 6 ) , w e have Lj - ipTj < (1 d^ < (-log ip)pj..
ty)p..
Hence,
Case 2:
(7)
Lj
< HJTJ <LJ
+ T\vj
+ 1
.
The r i g h t - s i d e i n e q u a l i t y of Eq. ( 7 ) , t h e obvious p- ^_Lj9 and t h e a s s u m p t i o n
P i .> •• •_> P n imply i)T. <. 1^- + ippj. + 1 _< Z^- + ipp. £ Lj + i j ^ . Hence,
T h i s and t h e l e f t - s i d e i n e q u a l i t y of (7) g i v e
1 - * £ ™f< *•
2V
Now we have jf(^) = 0 and
/ ( l - 40 = ( - l o g i|))(l + i|>) - fl(ip, 1 - *)
= ( - l o g i | 0 ( l + <|0 - ( - l o g ijj)(2 - l|0
=
(-log
I|J)(2I|> -
1).
T h e r e f o r e , by t h e downward c o n v e x i t y of f(x),
f ( x ) <_ ( - l o g ifOOP - # )
we have
if
1 - ty £ a; £ i|i.
Hence,
^- = ^(57)
i (- lo s *>(<^
-J>
- ^
But by the right-side inequality of ( 7 ) , w e have tyTj - Lj <.ipp.+ 1 .
dj < (-log 4 0 %
+1
£ (-log ^)p i + 1 .
Hence,
1982]
AN ENTROPY VIEW OF FIBONACCI TREES
173
In either case, we have
dd <. (-log i))pj9
since p. _> p.
j = 1, ..., n - 1,
. This finishes the proof.
For example, take the English alphabet including "space" (n = 27) with
letter-frequencies given in [7], If we construct a tree by ^-weight-balancing
for this source, we obtain #/(-log ip) = 5.885, C = 5.958.
Remarks: The above proof can be modified to prove the same inequalities (with
ip replaced by A) for the average cost of the tree built by A-weight-balancing
whenever 1/2 <_ A £ ip, i.e., 1 .£ c •<. 2.
If we impose an appropriate condition on p1,
prove the upper bound on C.
P.7+1
Theorem 2: If — —
..., p , we may somewhat im-
1
>_ y^, j = 1, ..., n - 1, then
C < log
, H i,p + YV
I(J(1 - p ) .
*«'
Proof: It is sufficient to show that for Case 1 in the proof of Theorem 1
we have dj £ (-log ty)tyPj , because we have shown dj <_ (-log ty)tyPj + 1 for Case 2.
From (6) and the assumption, we see that
J
«/
T
r
<7 + 1
tj
The downward convexity of /(or) and a direct numerical check show
±f
fix) < c-iog wnf-E-jjJ
* ^ * ± rri'
from which it follows that
(Lj \
Lj ~ Wj
dj = Tof\j7)
<• ( - l o § W
i - ^
<• ( - l o § *)*Pj-»
usin
§
< 6 >-
This completes the proof.
4.
The Case p n = ••• = p and Fibonacci Trees
r
i
r
n
In this section, we shall restrict ourselves to the special but important
case p, = ••• = pn, i.e., all terminal nodes have equal weight. Let us first
define the Fibonacci
tree of order k according to [7].
Let
(F0, F19 F29 F3, Fh, F5, ...) = (0, 1, 1, 2, 3, 5, . . . ) ,
174
AN ENTROPY VIEW OF FIBONACCI TREES
[May
be the Fibonacci sequence. The Fibonacci tree of order k has Fk terminal
nodes, and it is constructed as follows: If k = 1 or 2, the tree is simply the
"terminal" root only. If k _> 3, the left subtree is the Fibonacci tree of
order k - 1; and the right subtree is the Fibonacci tree of order k - 2.
Remark: The Fibonacci tree we assign order k is called in [7] the "Fibonacci
tree of order k - 1." We choose this indexing for its neatness in our argument.
Figure 1 is the Fibonacci tree of order 7.
FIGURE 1.
The Fibonacci
tree
of order
7
Lemma 2: The Fibonacci tree of order k9 k >_ 2, has Fk_± terminal nodes of cost
k - 2 and Fk_2 terminal nodes of cost k - 1.
Proof (Induction on k): Trivially true when k = 2. The Fibonacci tree of
order 3 obviously has one (= F2) terminal node of cost 1 and one (= F 2 ) terminal node of cost 2. Suppose the lemma is true for each Fibonacci tree of order less than k9 k >_ 4. By the construction of the Fibonacci tree of order k
it has, in the left subtree, Fv_0 terminal nodes of cost (fc-3) + l = - / c - 2
and F-,
1, and, in the right sub(k - 2) + 1 = k
k- 3 terminal nodes of cost
k - 2 and Fk_h terminal nodes
tree, • k - 3 terminal nodes of cost {k - h) + 2
of cost (k - 3) + 2 = k - 1. Hence, the Fibonacci tree of order k has, in all,
Fk_ + Fk_3 = Fk_1 terminal nodes of cost k - 2 and F-k_3 + Fk_
Fk_2 terminal nodes of cost k - 1. This completes the proof.
Theorem 3: The average cost of the Fibonacci tree of order k is given by
F
k-2
C = -4r^+ (k - 2).
Proof: By Lemma 2, we have
C = -^-{(fc - 2)Fk..±
Since
+ (k - l)Fk_2]
= i~{Fk_2
+ (k -
2)Fk}.
1982]
AN ENTROPY VIEW OF FIBONACCI TREES
Fk =j^>'k
we have Fk
175
" (-*>*}• by [6],
—if; ~k when k becomes large.
Therefore, for large k9
C-<(k-2)+^2=k-l-\l).
The following procedure, due to Varn [12], constructs an optimal tree (in
the sense of minimum average cost) for general c% Suppose an optimal tree with
n - 1 terminal modes has already been constructed. Split, in this tree, any
one terminal node of minimum cost to produce two new terminal nodes. The resulting tree with n terminal nodes will be optimal. The validity of this procedure is an immediate consequence of Lemma 1:
The left-hand side is the average cost to be minimized when p x =
minimize the left-hand side is to minimize the sum of costs of n
nodes; i.e., to minimize
•-p„.
To
1 internal
n-i
J=I
Consider the infinite complete binary tree, and use the "greedy" procedure to
pick the n - 1 cheapest nodes to be internal. It is easy to see that this
grows a tree optimal at each step, the same tree as grown by Varn's procedure.
Returning to our case o = 2, we have the following:
Theorem 4:
The Fibonacci tree of order k is optimal for each k _> 2.
Proof: From Lemma 2, the Fibonacci tree of order k >_ 2 has Fk_1 terminal
nodes of cost k - 2 and Fk_2 terminal nodes of cost k - 1. Hence, by VarnTs
procedure, it is sufficient to prove that if we split all terminal nodes of
cost k - 2, then the resulting tree, which then has Fk_2+ 2Fk_-L = Fk^1 + Fk =
Fk+1 terminal nodes, is the Fibonacci tree of order k + 1. To prove this by
induction on k9 suppose the assertion is true for the Fibonacci trees of order
less than k9 k >_ 3. (When k = 2, the assertion is trivially true.) The left
subtree of the Fibonacci tree of order k is the Fibonacci tree of order k - 1
with Fk_2 terminal nodes of cost (k - 3) + 1. Splitting these nodes produces
the Fibonacci tree of order k by the induction hypothesis. Similarly, the
right subtree is the Fibonacci tree of order k - 2 with Fk_3 terminal nodes of
cost (k - 4) + 2. Splitting these nodes produces the Fibonacci tree of order
k - 1 by the induction hypothesis. Therefore, splitting all terminal nodes of
cost k - 2 of. the Fibonacci tree of order k produces the Fibonacci tree of
order k + 1.
Theorem 5: Express the number of terminal nodes by n = Fk + r for some k >_ 2
and 0 £ r < Fk_1.
The tree built according to ip-weight-balancing is optimal,
with the average cost given by
[May
AN ENTROPY VIEW OF FIBONACCI TREES
176
F
+ 3r
When r = 0, the tree is the Fibonacci tree of order k.
Proof: When k = 2 or 3, the theorem is trivially true. Suppose k >. 4. We
prove by induction on the number of terminal nodes that the tree having Fk + r
terminal nodes and built by ip-weight-balancing is the same as a tree constructed from the Fibonacci tree of order k by splitting r of it's Fk_1 terminal nodes of cost k - 2. Varnfs procedure and Theorem 4, then, prove the optimality part of the theorem. When n - 3, we have k - 4 and r = 0. So the
assertion is true, since the^ ijj-weight-balancing for n - 3 produces the Fibonacci tree of order 4. Suppose the assertion is true for each number of terminal nodes less than n ~ Fk + r9 & _> 4. By Lemma 2 and the construction of
Fibonacci trees, there are Fk_2 terminal nodes of cost k - 2 in the left subtree of the Fibonacci tree of order k9 and there are Fk_3 terminal nodes of
cost k - 2 in the right subtree. Hence, we need only show that the ifj-weightbalancing "divides" Fk + r into Fk_1 + s, 0 <_ s < ^ . 2 for the left subtree,
and Fk_2 + t9. 0 <_ t <_ Fk_3 for the right subtree, with s + t - r.
If this is
true, then we can apply the induction hypothesis and incorporate the cost of
the initial branch to find that the tree built by ip-weight-balancing on the
left is obtained by splitting s of its terminal nodes of cost (k - 3) + 1 and
that on the right by splitting t of its terminal nodes of cost (k - 4) + 2.
Let us show, therefore, for the left, that the integer m given by
m - (1 - t|0 < ]\)(Fk + r)
corresponding to (5), satisfies m - Fk_1
<_ m + ip,
+ s, 0 £ s < Fk_2.
Using
the above inequalities may be written as
F
k-i
+ * p ~ ^ ~ (~Wk
1 m < Fk_x
+ tyr - ty• - (-ty)k
+ 1.
Since (-i|j)k < tjj2 = 1 - ty9 we have
-1 < _ \p - (-^
<_ ^r - y - {-^)k ,
and, on the other hand,
tyr - ty - (-\p)k + 1 <_ ^(Fk_1 - 1) - i\) - H > ) k + 1
= ^.
2
- ( - ^ ) k _ 1 - * - (-^) fe + ^
= Ffc_2 - ^
2
- (-^) fe } < F f e _ 2 .
Therefore, Fkmml - 1 < m < Fk_± + Fk_2; thus, m = Fk_1+s
0 <_ s < Fk_2.
Similarly, we can show, for the right, using Fk_2
that the integer m given by
for some s such that
- (1 - ty)Fk = -(-ip)* ,
1982]
AN ENTROPY VIEW OF FIBONACCI TREES
177
m - ty < (1 - i\)){Fk + p) < m + (1 - \p)
satisfies m = Fk_2 + t, 0 £ t <_ Fk_3.
The average cost, then, using Lemma 2, is given by
c
=
F
\
r
{ ( k
~ 2><F*-i -*> + < * - D ( ^ - 2 + r) + kr}
+ 3r
F
= JirL-z
+ (fc - 2).
This completes the proof.
The height
of a tree is defined as its maximum level, the length (the number of branches) of the longest path from the root to a terminal node. A binary tree is called balanced
(the concept due to Adelson-Velskii and Landis
[1]) if the height of the left subtree of every internal node never differs by
more than 1 from the height of its right subtree.
Theorem 6: When p
= • • • = pn , the tree built by ip-weight-balancing is balanced.
Proof: Let the number of terminal nodes be Fk + r9 0 £ r < Fk_1.
It is
easily seen, by induction on k9
that the Fibonacci tree of order k >_ 2 is of
height k - 2 and hence balanced, and, if k _> 4, has only the leftmost two terminal nodes at the maximum level, with cost k - 2 (for the left node) and k 1 (for the right node). From the proof of Theorem 5, the ijj-weight-balanced
tree having Fk +'r terminal nodes is made by splitting r = s + £ (0 ± s <
Fk_2)
terminal nodes of cost k - 2 of the Fibonacci tree of order k with s from the
left subtree and t from the right subtree. In this splitting process, the
leftmost terminal node at the maximum level is, however, never split as long
as 0 £ r < Fk_19
and the tree remains balanced. Since s < Fk_2i
t <_ Fk_39
this assertion is readily seen by induction.
Theorem 7: For p1 = • • • = pn and an arbitrarily given number of terminal nodes
(or branch nodes), the tree built by ip-weight-balancing is the highest of all
balanced trees.
Proof: It is easily seen by induction on height that the balanced tree of
height h with a minimum number of terminal nodes is the Fibonacci tree of order h + 2 [8], Now suppose that there exists a balanced tree of height h with
Fk + v (0 <_ p < Fk_1)
terminal nodes, then
** + 2 <^k+r<Fk+
F ^
= Fk +
19
hence, h <. k - 2. But from the proof of Theorem 6 we know that the ip-weightbalanced tree on Fk + r nodes has height k - 2.
A Hypothetical Class of "Natural Trees"
It is amusing to draw (suggested by [10]) the Fibonacci trees upside down
so that they look like real trees or shrubs, with each branch of cost 2 about
178
AN ENTROPY VIEW OF FIBONACCI TREES
[May 1982]
twice (relatively) as long as its brother branch of cost 1 (i.e., bifurcation
ratio 1:2; asparagus, as the author observed, seems to grow in this way).
There may be variations in drawing. Figure 2 is a corresponding sketch of the
Fibonacci tree of order 7 shown in Figure 1. As we saw in the last section,
the simple repeating pattern (Fibonacci recursive rule) in the Fibonacci tree
implies, and is implied by, the entropic balancing of the tree. This, along
with the properties given in Theorems 5 and 7, might be of morphological interest for a class of mathematical "natural trees."
FIGURE 2.
Sketch
of a "natural
tree"
References
1. G. M. Adelson-Velskii & E. M. Landis. "An Algorithm for the Organization
of Information." Soviet Math. (Doklady) 3 (1962):1259-1263.
2. I. Csiszar. "Simple Proofs of Some Theorems on Noiseless Channels." Information
and Control
14 (1969);285-298.
3. R. M. Fano. The Transmission
of Information.
Tech. Rep. No. 65, M.I.T.,
March 1949.
4. Y. Horibe. "An Improved Bound for a Weight-Balanced Tree."
Information
and Control 34 (1977):148-151.
5. Y. Horibe. "A Note on the Cost Bound of a Binary Search Tree." J. Combinatorics3 Information
& System Sciences
5 (1980):36-37.
6. D. Knuth. Fundamental Algorithms.
Boston: Addison-Wesley, 1968.
7. D. Knuth. Sorting
and Searching.
Boston: Addison-Wesley, 1973.
8. K. Mehlhorn. "Searching, Sorting, and Information Theory."
Mathematical
Foundations
of Computer Science
1979. Ed. by J. Becvar. Berlin: Springer
Verlag, 1979, pp. 131-145.
9. C. E. Shannon. "A Mathematical Theory of Communication." Bell Syst.
Tech.
J. 27 (1948):379-423, §9.
10. H. Steinhaus. Mathematical
Snapshots.
Oxford: Oxford Univ. Press, 1969.
11. F. Topstye. Informationstheorie,
eine Einfuhrung.
Stuttgart: Teubner, 1974.
12. B. Varn. "Optimal Variable Length Codes: Arbitrary Symbol Cost and Equal
Code Word Probability." Information
and Control
19 (1971):289-301.
*****
ELEMENTARY PROBLEMS AND SOLUTIONS
Edited
by
A. P. HILLMAN
of New Mexico,, Albuquerque,
University
NM 87131
Send all communications
regarding
ELEMENTARY PROBLEMS AND SOLUTIONS to
PROFESSOR A. P. HILLMAN, 709 SOLANO DR., S.E.;
ALBUQUERQUE, NM 87108.
Each
problem or solution
should be on a separate
sheet (or sheets).
Preference
will
be given to those that are typed with double spacing in the format used
below.
Solutions
should be received
within four months of the publication
date.
DEFINITIONS
The Fibonacci numbers Fy, and Lucas numb eirs Ln satisfy
F
and
E
n+2
F
+ F
L
+
n +1
F
= 0
F
^ ' LQ = 29 L1
= 1
=
I.
L
Also, a and (3 designaten+2the roots (1 + /5)/2 and (1 - /5)/2, respectivelys of
x2 - x - 1 = 0.
PROBLEMS PROPOSED IN THIS ISSUE
B-472
Proposed
by Gerald
E. Bergumr
Find a sequence {Tn}
currence Tn = ccTn x + bTn 2
{Tn}.
B-473
Proposed
by Philip
Proposed
by Philip
State
University,
Brookings,
SD
satisfying a second-order linear homogeneous resuch that every even perfect number is a term in
L. Maria, Albuquerque,
Let a = Lxooos ^ = ^iooi» °
a factor of 1 + xa + xh 4- x ° + xd1
B-474
S. Dakota
==
NM
^ioo2> d = £1003Explain.
L. Mana, Albuquerque,
Is 1 + x +. x1 + x* •+xh
NM
Are there an infinite number of positive integers n such that
Ln + 1 = 0 (mod
2n)l
Explain.
B-475
Proposed
by Herta
T. Freitag,
Roanoke,
VA
n
Let Sk(n)
=
E
(-1)J'+ V /C .
Prove that \S3(n)
- S\{n)\
is 2[(n + l)/2]
j=i
times a triangular number.
Here [ ] denotes the greatest integer function.
179
B-476
[May
ELEMENTARY PROBLEMS AND SOLUTIONS
180
Proposed
by Herta
T. Freitag,
Roanoke,
VA
n
Let Sk (n) = £
(~iy'
+1 k
Prove that \Sn(n) + Sz(n)\
j.
is twice the square
j-i
of a triangular number.
B-477
Proposed
by Paul S. Bruckman,
Sacramento,
CA
(-l)n
1
1
Prove that Y* Arc tan —
= — Arc tan —.
F
n-2
2
2n
2
SOLUTIONS
Casting Out 27's
B-446
Proposed
by Jerry
M. Metzger,
University
of N. Dakota,
Grand Forks,
ND
It is familiar that a positive integer n is divisible by 3 if and only
if the sum of its digits is divisible by 3. The same is true for 9. For 27,
this is false since, for example, 27 divides 1 + 8 + 9 + 9 but does not divide
1899. However, 27|1998.
Prove that 27 divides the sum of the digits of n if and only if 27 divides one of the integers formed by permuting the digits of n.
Solution
by Paul S. Bruckman,
Concord,
CA
Given
m
(1)
N =
Y,aklOk,
fc = 0
where the ak
s are decimal digits, let the sum of the digits be given by
m
(2)
8(210 = £
ak.
k=o
We begin by observing that the statement of the problem is false. The correct
statement should read as follows: If 27 divides the sum of the digits of N9
then 27 divides one of the integers formed by permuting the digits of N. The
converse is clearly false, since, e.g., 27| 27 but 27|s(27) = 9.
Suppose
(3)
27|s(il/).
The smallest positive integer N satisfying (3) is 999. Since 27|999, we see
that the (modified) proposition is verified for N = 999. We may therefore suppose m _> 3.
Since 9\s(N)9
thus 9\N. Let (PN denote the set of all possible integers
M formed by permuting the digits of N. Since s(M) = s(N)
for all M e (PN 9 we
see that 9\M for all M e &N. We will assume that M = ±9 (mod 27) for all M e
(PN and show that this leads to a contradiction.
Given k (0 <. k <_ m - 2) , form /17/c(1) e (PN and N£2) e <PN by merely permuting the triple (ak9 ak+19 ak+2)
to (ak+l9
ak+2,
ak) and (ak+29
ak9 a^+i), respectively. Then
1982]
ELEMENTARY PROBLEMS AND SOLUTIONS
+ 10fc+1(afc+2 - afc+1) + 10 k+2 (a, -
- N - 10*<afc+1 - ak)
N™
E
l° k ta k+1 - ak + 10(a k+2 - ak+1)
k
= -9 • I0 (ak
+ ak+1
+ ak+2)
= -9(ak + afe+ 1 + ^^+2)
+ 19(ak - ak+2)}
181
ak+2)
(mod 27)
(mod 27)
(mod 27).
Similarly, we find that
tf£2) - N = 9(ak
+ ak+1 + ak+2)
(mod 27).
Having assumed that N = ±9 (mod 27), we cannot have
a
k
+
a
+
k +l
a
k+2
± l
~
(mod
3
) '
for we would then have either
N^1)
= 0
contradicting the assumption.
(4)
ak + ak+1
+ ak+2
il/£2) E 0
or
(mod 27),
Since k is arbitrary, we must therefore have
= 0
(mod 3), k = 0, 1, ..., m - 2.
Thus the sum of any three consecutive digits of N must be divisible by 3. But
we see by symmetry that this same property must hold for all M e (P' . This can
only be true if all the ak's are congruent (mod 3).
Suppose, therefore, that
(5)
ak = 3bk + r, where bk = 0, 1, 2, or 3,
v = 0, 1, or 2,
with
bk = 3 only if p = 0.
Let
m
(6)
5 - Z
bk!0k.
k =0
m
Then N = £
k =0
m
(32>k + r ) 1 0 / c = 3£ + r £ 1 0 k , or
fe
=0
' fl = 3B + | ( 1 0 m + 1 - 1 ) .
(7)
Also,
m
s(N)
(8)
s(N)
We consider two (a pvlovi)
= E Ohk + r), or
.fc-o
= 3s (5) + r(m" + 1 ) .
possibilities:
(a) m E Q or 1 (mod 3). Since 3\s(N),
717 = 35 and s (N) = 3s (B). But
we see from
27|e(i\^) —>.9|s(B) — > 9 | 5 . —
(8) that r = 0.
Hence,
182
ELEMENTARY PROBLEMS AND SOLUTIONS
which contradicts our assumption.
(b) m = 2 (mod 3).
[May
This leaves the only remaining possibility:
Let m = 3t - 1, t >_ 2.
Note that
|(10 m +1 - 1) = £(103* - 1) = p 3 £ 1 1 0 k = lllr £ 103*
^
fc
=0
fc
=0
= 3r k£= 0x (mod 2 7 >E 3 ^ (mod 27 >-
Thus, using (7),
N = 3B + 3rf- (mod 27) — > | = B + rt
Also, ^ p - = s(B)
+ rt
E B + rt
(mod 9).
(mod 9).
Hence, | E ^ p - (mod 9), which, to-
gether with 271s(N) implies 27|#, again contradicting our assumption.
Thus the assumption is false, establishing the (modified) proposition.
Also
solved
by the
proposer.
Casting Out Eights
B-447
Based on the previous
proposal.
Is there an analogue of B-446 in base 5?
Solution
by Paul S. Bruckman,
Concord,
CA
Given
m
(1)
ak5k,
N = £
fc = 0
where the ak's
are digits in base 5 (ak = 0, 1, 2, 3, or 4) , let the sum of the
digits be given by
m
(2)
8 0)
m
We n o t e t h a t N - s(N)
(3)
= £
ak.
k=o
•
= £ ak(5k - 1) '= 0 (mod 4 ) , s i n c e 5k = 1 (mod 4 ) . Thus
fc = o
4|iV i f f
4|s(71/).
The analogue suggested by B-446 would probably read as follows: If 8 divides
the sum of the digits of N (in base 5), then 8 divides one of the integers
formed by permuting the digits of N (in base 5).
Unfortunately, the above proposition is false, unless additional conditions on N are specified. A counterexample is
N = 3,908 = (111113)5;
although S\s(N) = 8, we find that all six integers formed by permuting the
digits (in base 5) of N are congruent to 4 (mod 8 ) , and therefore not divisible by 8.
ELEMENTARY PROBLEMS AND SOLUTIONS
1982]
183
The following is the corrected (though more complicated) version of the
proposition: If 8 divides the sum of the digits of N (in base 5), then 8 divides one of the integers formed by permuting the digits of N (in base 5 ) ,
unless
all the digits of N are odd (i.e., 1 or 3) and the number of such digits
is congruent to 2 (mod 4). In the latter case, all the permutations are congruent to 4 (mod 8).
(The proof is similar
lems
Editor.)
to that of B-446 and was deleted by the Elementary
Prob-
Sum of Products Modulo 5
B-448 Proposed by Herta T. Freitag,
Roanoke, Va
Prove that, for all positive integers t ,
u
i =l
Solution
by John Ivie,
Glendale, AZ
It suffices to show that each pair F5^+1L5i
+ ^si+e^5i+5i^n
tion is divisible by 5. Using the Binet Formula, this pair equals
T?
10Ul
r
4- F
= ST
J1J
^lOi+ll
10i
t n e sulrana
~
+ Sm
Also solved by Pauls. Bruckman, Bob Prielipp,
Charles B. Shields, Sahib Singh,
Lawrence Somer, Charles R. Wall, Stephen Worotynec, Gregory Wulczyn, and the
proposer.
Sum of Products Modulo 7
B-449 Proposed by Herta T. Freitag,
Roanoke, VA
Prove that, for all positive integers t,
£(-Di + ^ 8i +1 £ 8 i = 0 (mod 7).
i =1
Solution
by Charles R. Wall, Trident
Technical
College,
Charleston,
SC
It is easy to show that
The powers of -1 telescope, since the number of summands is even, and thus
E (-ir
i* 1
We group the It
visible by 21:
+1
F,i + 1Lai = E(-l) i+ 1 ^ e i + 1t=i
summands in the latter sum into t pairs, each of which is di-
[May 1982]
ELEMENTARY PROBLEMS AND SOLUTIONS
184
The stronger result, that the original sum is divisible by 21 (rather than
merely 7), follows at once.
Also solved by Pauls. Bruckman, John Ivle,
Somer, Stephen Worotynec,
Gregory Wulczyn,
Bob Prielipp,
Sahib
and the
proposer.
Singh,
Lawrence
Lucas Quadratic Residue
B-450
Proposed
by Lawrence
Somer,
Let the sequence {Hn}n=Q
Washington,
B.C.
be defined by Hn = F2n
+
F2n+2.
(a) Show that 5 is a quadratic residue modulo Hn for n >_ 0.
(b) Does Hn satisfy a recursion relation of the form Hn + 2 = eHn + 1 + dEn, with
c and d constants? If so, what is the relation?
Solution
by E. Primrose,
University
of Leicester,
England
We prove (b) first, and use it to prove (a).
(b)
Examination of the first few terms suggests that
=
Hn+2
3Hn+1
-
Eny
and this is easily verified by using the defining relation for Hn and the recurrence relat ion ror r^.
(a) We prove that H^+1 - HnHn+2
H
n+1
"
= 5, which gives the required result.
H H
=
n n+2
=
2
It follows by induction that H
n+1
H
3
n+1
" #n( #n+l
Hn + Hn+1(Hn+1
- HnHn+2
H
"
Now
n)
- 3Hn)
= Hn -
Hn_1Hn+1.
= H\ - HQH2 = 5.
Also solved by Paul 5. Bruckman, Herta T. Freitag,
John Ivie,
John W: Milsom,
Sahib Singh, Bob Prielipp,
A.G. Shannon, Charles R. Wall, Gregory Wulczyn,
and
the
proposer.
Consequence of the Euler-Fermat Theorem
B-451
Proposed
by Keats
A. Pullen,
Jr.,
Kingsville,
MD
Let k9 m9 and p be positive integers with p an odd prime. Show that in
base 2p the units digits of wfc(>P~1)+:L ±s the same as the units digit of m.
Solution
by Charles
R. Wall,
Trident
Technical
College,
Charleston,
SC
Let (J) be Euler's function. Since p is an odd prime, <|>(2p) = p - 1 and
therefore by Euler's Theorem, mp~1 = 1 (mod 2p). We take the fcth power of both
sides and then multiply both sides by m to obtain
=
muP-D+i
as asserted.
[This assumes gcd(m,
Also solved by Paul S. Bruckman,
Lawrence Somer, Gregory Wulczyn,
m
(mod
2p)
p) = 1 but also follows for p|m.]
Herta T. Freitag,
Bob Prielipp,
and the
proposer.
*****
Sahib
Singh,
ADVANCED PROBLEMS AND SOLUTIONS
Edited
by
RAYMOND E. WHITNEY
Lock Haven State
College,
Lock Haven, PA 17745
Send all communications concerning ADVANCED PROBLEMS AND SOLUTIONS to:
RAYMOND E. WHITNEY, MATHEMATICS DEPARTMENT, LOCK HAVEN STATE COLLEGE,
HAVEN, PA 17745.
This department especially
welcomes problems believed
to be
new or extensions
of old results.
Proposers should submit solutions
or other
information
that will assist
the editor.
To facilitate
their
consideration,
solutions
should be submitted on separate signed sheets within 2 months
after
publication
of the problems.
PROBLEMS
H-339
Proposed by Charles
R. Wall,
Trident
Technical
College,
Charleston,
CA
A dyadic rational is a proper fraction whose denominator is a power of
2. Prove that 1/4 and 3/4 are the only dyadic rationals in the classical
Cantor ternary set of numbers representable in base three using only 0 and 2
as digits.
H-340
Proposed by Verner E. Hoggatt,
Jr.
(Deceased.)
Let A2 = B9 Ah = C9 and A2n + i* = A2n ~ A2n
H-341
= (~l)n
+1
(n = 1, 2, 3, . . . ) .
a.
A2n
b.
If A2n > 0 for all n > 05 then B/C = (1 + /5)/2.
(Fn_2B
-
+2
Show:
F^C).
Proposed by Paul S. Bruckman, Corcord, CA
Find the real roots, in exact radicals, of the polynomial equation
(1)
p(x)
Ex
6
-
4x5 + lxh
- 9x3 + 7x2 - kx + 1 = 0.
SOLUTIONS
Once Again
Professor M. S. Klamkin has pointed out that this problem was proposed
previously by him (Amer. Math. Monthly 59 (1952):471]. It also appears in an
article by W. E. Briggs, S. Chowla, A. J. Kempner, and W. E. Mientka entitled
M
0n Some Infinite Series," Soripta
Math. 21 (1955):28-30.
185
LOCK
186
ADVANCED PROBLEMS AND SOLUTIONS
H-320
[May
Proposed by Paul S. Bruckman, Concord CA
(Vol. 18, No. 4, December 1980)
Let
oo
C(s) = 2 n " s » Re(s) > 1> the Riemann Zeta function.
M-l
Also, let
n
Hn = X ) k"1, w = 1, 2, 3, ..., the harmonic sequence.
fc -1
Show that
% = 25(3).
t
n - l ft
Solution
by C. Georghiou,
University
of Patras,
Patras,
Greece
Method I: Clearly, the series converges; let S denote its sum.
K
- ±k-> - £ ( £ - j ^ ) --twTTn)
- -njy^a
We note that
-t)dy
and
~
x3~1e'xdx
If*
Us;
»-i-
1 - e~x
^o
where T(s) is the Gamma function.
Then
*-£%--£: C^lo*\-»dt
n - i n2
.
/
Jo
n=ii0 n
l o g 2 ( l - *), ^
V
••fi(qio«v^
*
;0
£
s i n c e
n=
iVn /
£
l^-..iog(l - t) for |*[ < 1,
n-ln
and the interchange of the summation and integration signs is permissible.
Setting t = 1 - e~x in the last integral, we get
~~2e-xdx
-co
/ • • £
e
-/0
'o 11
^
-a; - 2C(3).
"5
Method 11: Clearly, the series converges; let S denote its sum.
0°
n
Hn =
Define a l s o
#n - E k " 2
k= l
Then
/ -.
-I
£y=k?^
\
We note that
oo
=
• F + ^ ) £ k(fc+«)and # 2 = ^ ( 2 ) - E2n.
-
2
Hn = jT,
, the series 22 — converges, and we denote its sum by S.
fc-i (k +
ft)2
»-i n
1982]
ADVANCED PROBLEMS AND SOLUTIONS
The following hold true:
(i)
E
- S.
n=1
(n + l ) 2
Indeed, by rearranging the above series, we get
oc
2^
n =1
°°
"-n
° ° 1
-j
°°
-.
°°
.
(ii)
Hk
oo
- = 2^
— 2^ i7 = 2^ i7 2^
(n + 1)
n-i (n 4- l) 2 fc = i K
*-i ;< » - i ( n +
~ ~ 2-# "ir
fc) * = 1
_
=
S;
5 = C(3) + ~S.
I n d e e d , w i t h H. = 0,
oo
—
+
#
,
nz
n=i
O
O
H
oo
T
n = i nz
n= ind
#
oo
^
«-i (n + 1).
which, by means of (i) establishes (ii).
(iii)
IS = 5 .
Indeed,
_
S
oo
Hn
co
oo
,
°°
°°
1
1
/
7>
\
I/
1
\
I
" ,?i ~ " S, £„(„ + *>* " «?. £ JFWT^) " ^ T ^ l ) j
from which (iii) follows.
Combining (ii) and (iii), we have S = 2^(3).
Also
solved
by L. Carlitz,
A. G. Shannon,
and the
proposer.
Big Deal
H-321
Proposed by Gregory Wulczyn, Bucknell
(Vol. 18, No. 4, December 1980)
University,
Lewisburg,
PA
Establish the identity
C1.W
+
F
n
~ <£l2r
+
L
e, + L,r
" » ^Liz*
+
F
L
2,
>
+ (£20r + L16r + ^ r + 3 ) ( ^ + 1 0 r + F%n +
~ (i 2 „ r -
L
20r
= 40(-l)"n Ani =±
+ L
12r
+
2L
Sr
" ^ ( ^ n + sr
hr)
+
F
« + 6r)
188
Solution
ADVANCED PROBLEMS AND SOLUTIONS
by Paul S. Bruckman,
[May
Concord, CA
Let
(1)
<Kn, r ) = Fsn + 1,r
+ F*.- Ar(F6n+12r
+ F* + 2r) + B (F* + 1 0 r + Fsn + ,r)
— r'
(F6
+
where
(2)
A r = L12r + L 8 r + L 4 r - l r
(3)
Bv - L2Qr
(4)
C;p = Z/ 21+r - L 2 0 r + Ir 122 »
6
F-6r
^
+ -^i6r ~*~ ^ 4 r ~*~ 3 ;
+
2
^8r ~ 1.
We make repeated use of the following identities:
(5)
=
^2u^2v
^2u + 2v
+
^2u-2v
»
Fm6 = 5 ' 3 { £ 6 m - e C - l f L ^
(6)
I t i s a tedious but straightforward
t i e s , by means of ( 5 ) :
L
(7)
mkr
~ ArL1Qkr
+ BrLskr
+ 15L2m - 2 0 ( - l ) m } .
e x e r c i s e t o prove t h e following
- CrL2kr
identi-
= 0, k = 1, 2 , 3 .
Let
(8)
Ur = Cr - Br + Av - 1.
We n o t e t h a t
l25F22rFlFl
- V*r ~ 2 ) ( £ 8 r " 2 ) ( L 1 2 , - 2)
= \Lkr - 2 ) ( L 2 0 r + -^^r ~ 2 L 1 2 r - 2L 8 r + 4)
= L2i+r - 2 L 2 0 r - Lier
(after
+ 2L 122 , + 3LQr
- 6
simplification)
= O^r
- £ 2 0r + £12*
+
2L
*r
X
) ~ ( L 20r + ^ISr + ^ r
"
+ ^12,
+
^r
+ £,,
+ 3>
- 1) - 1
or
Z7 = 125F 2 F 2 F 2 .
(9)
Now
125<|>(n, 3?) = £ 6 n+84r
" 6 (-1) n L 4 n + 5
+ £'6n
6 r
-
6(-l)"Li+n
+ 15£2n+28r
- 40(-l)
-
i4 r {L 6 n +72r
+
-^6n+12r
~ 6(-l)
Li+n + i+83" -
6(-l)
n
L^n+Qr
+ 1 5 L 2 n + 2 ^ + 15L2n+ ,r - 40(-l)n}
+ 15L 2n+20 P + 15L 2 n + 8 r - 40(-l)n}
+
15L 2 n
1982]
ADVANCED PROBLEMS AND SOLUTIONS
C
+
r^Sn+k8r
L
6n+36r
~
+
[using (6)]
^6n+i+2r ^ 4 2r
^ r ^ 3 0r
1 5
+
5
6
(-l)
L
i+n+32r
^ + 16, +
r^l8r
^r^er^
6 ( 1 ) ^ l + n + 2 8i» (L28r
-
"
15
189
6
( _ 1 ) " ^ 4 * +2 i ^
^ n + l 2 , " *0(-l)»}
"" ^ 2 - ^ 2 Or
+
B L
r l2r
" ^V^r )
+ 40(-l)n(-l + 4 r - 3 r + C r )
[using (5) repeatedly once again, and factoring]
= 40(-l)n[/2,
[using (7) and (8)], or as a result of (9),
<|>(n, r) = 4 0 ( - l ) n F ^ F ^ F ^ .
(10)
Also solved
by the
Q.E.D.
proposer.
Two Much
H-322 Proposed by Andreas N. Phillppou,
(Vol. 19, No. 1, February 1981)
American Univ. of Beirut,
Lebanon
For each fixed integer k >_ 2, define the ^-Fibonacci sequence f
n KK.J
Jo
and
Jn
fJ ik\
(a) /„(k) = 2r
(b) f n
i f 2 £ n £ fe,
if n >. k + 1.
+ •- + ^
n-1
Show the following:
(H
= 0, f™ = l ,
(
[ Jfn-1
^ ^+ ••• + f„ (fe)
(k)
by
r
< 2
if 2 <. n < k + 1;
if n >_ k + 2;
(c) E (/f)/2n) - 2*-1.
Solution
by the proposer
For 2 _< n _< /c,
j?(k)
_
2n-2f(k)
2»'*
and /<*"> - /<*> +
(fe+1) "*
+ f™ - 2f (*>
,fc-i
which establish (a). Next, for n >_ k + 1,
/».00
=
.<*)
? (fc)
fn-i
+ ••• + /n.fc
and
,(*>
(fc)
.(*)
/ n - 1 - fn_2 + • • • + fn
so that
(1)
,(*0 = 0Ak) __ „(fc)
Jn
A
J n-l
Jn-l-
k
(n >_ k + 1),
,(k)
Taking n = k + 2 in (1), (a) implies /fcv+2 = 2^ - 1 < 2k, which verifies (b)
for n = k + 2. Assume now '/ m ^ < 2 m for some integer m (>_ k + 3). It then
[May
ADVANCED PROBLEMS AND SOLUTIONS
190
follows by (1) and the positivity of ff^
z
Jm+l
im
(i J> 1), that
Jm-k
L
9
and this proves (b). Using (1) again, we get
(ff ) /2 n ) - Cf<*)1/2B+1) = (/„(.\)/2B+1) > 0
( n > k + 1).
Therefore,
11m (ff} /2") = 0.
(2)
n + oo
"
m
Setting s^
algebra,
= /J (/n
n=1
/2n) (m ^L 1) , and using (1) and (a), we get, after some
8<« = 2*'1 - (2k+1 - l)(/f ) /2 m ) + £ (f,f)i/2'"-i)
(3)
(m >.fc+ 2).
i =l
= 2 ~ 1 , and this shows (c).
Relations (2) and (3) give lim s^
m -> oo
Remark 1: For fc = 2, (b) reduces to Fn < 2 n " 2 if n _> 4.
and Scott [4] proved Fn < 2n~2 if n >_ 5).
Remark 2:
(Fuchs [2] proposed
For k = 2, (c) reduces to 2(^n/2 n ) = 2, a result obtained by Lind
n-l
[3] in order to solve a problem of Brown [1],
References
1.
2.
3.
4.
J. L. Brown. Problem B-118. The Fibonacci
Quarterly
5, no. 3 (1967):287.
J. A. Fuchs. Problem B-39. The Fibonacci
Quarterly
2, no. 2 (1964): 154.
D. Lind. Solution of Problem B-118. The Fibonacci
Quarterly
6, no. 2
(1968):186.
B. Scott. Solution of Problem B-39. The Fibonacci
Quarterly
2, no. 3
(1964):327.
Also
solved
by P. Bruckman and L.
Somer.
A Common Recurrence
H-323
Proposed by Paul Bruckman,
(Vol. 19, No. 1, February
Concord,
1981)
CA
Let (con)°^ and Q/ M )^ be two sequences satisfying the common recurrence
(1)
p(E)zn
= 0,
where p is a monic polynomial of degree 2, and E = 1 + A is the unit right-shift
operator of finite difference theory. Show that
(2)
xnyn+i
- xn+iyn
=
(p(o)) n (^ 0 ^i " x i^ 0 )> n = o, i , 2, ... .
Generalize to the case where p is of degree e >_ 1.
Solution
by the
proposer.
We solve the general case, with p any monic polynomial of degree e >.!.
Suppose
(„U) }°° f„(2) \°°
/(e) \°°
\<»n
I 0'
\^n
i 0»
•••s
\<>
n
Jo
1982]
ADVANCED PROBLEMS AND SOLUTIONS
191
are sequences satisfying the common recursion (1). We seek to evaluate Casorati's Determinant
(3)
,<D
,(2)
,(*>
,(D
,(2)
,<*>
„(2)
^n+e-i
,(«>
Dn
~U)
n+e-1
Let Un = {(zji\-i))e*e
be the matrix whose determinant is Dn; also, de-
fine the e X e matrix J as follows:
(4)
0
0
0
1
0
0
0
1
0
0
0
1
J =
0
-p(0)
0
0
-p'(0)/l!
-p"(0)/2!
0
...
1
-p'"(0)/3! ... -p (e _ 1 ) (0) / (e - 1) !
Note that p has the Maclaurin Series expansion
po-t^Sia...
(5)
r=0
Therefore, the sequences (s„ ) 0 (& = 1» 2, ..., e ) satisfy the common recursion
P=O
*"
? =o
since p is monic, p (e) (0)/e! = 1, and hence
(6)
L«I
p?
*r,
(n = 0, 1, 2, . . . ) .
We now observe, using (3), (4), and (6), that
(7)
J • Un = Z7 n+1 , n = 0, 1, 2, ... .
It follows by an easy induction that
(8)
Un = JnUQS
n = 0, 1, 2, ... .
We may evaluate \j\ along the first column of J, and we find readily
that \j\ = (-l)e_1(-p(0)) = (-l)ep(0). Therefore, taking determinants in (8)
yields:
Wn
£o>
192
ADVANCED PROBLEMS AND SOLUTIONS
[May 1982]
Dn - {(-l)ep(O)}nZ)0, n = 0, 1, 2, ... .
(9)
This is the desired generalization of (2). Many interesting identities
arise by specializing further. For example, taking
p(z)
= z2 - z - 1, (xn)
= (Fn)9
and (z/„) = (L„) ,
yields:
(10)
F n I n + 1 - Fn+1Ln
= 2C-1)"" 1 , n - 0, 1, 2
*****
ERRATA
In the article "On the Fibonacci Numbers Minus One" by G. Geldenhuys,
Volume 19, no. 5, the following two errors appear on pages 456 and 457:
1.
The recurrence relation (1), which appears as
D1 = 1 + y, D2 = (1 - y ) 2 , and Dn = (1 + y ) ^ . ! - \iDn_3
for n >_ 3
should read
D± = 1 + y, Z?2 = (1 + y ) 2 , and Dn = (1 + u)Z?n-1 - y£>„_3 for n _> 3;
2.
The alternative recurrence relation (4), which appears as
Vm -
0»-i "
^ - 2 = 1 for m > 3
should read
Vm " V^m-i - ^ m - 2 = 1 f or m' >. 3.
We thank Professor Geldenhuys for bringing this to our attention.
*****
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Clyde Bridger
J.L. Brown, Jr.
P.S. Bruckman
P.F. Byrd .
L. Carlitz
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UNIVERSITY OF SCRANTON
Scranton, Pennsylvania
UNIVERSITY OF TORONTO
Toronto, Canada
WASHINGTON STATE UNIVERSITY
Pullman, Washington
BOOKS AVAILABLE THROUGH THE FIBONACCI ASSOCIATION
Introduction to Fibonacci Discovery by Brother Alfred Brousseau. Fibonacci
Association (FA), 1965.
Fibonacci and Lucas Numbers by Veraer E. Hoggatt, Jr. FA, 1972.
A Primer For the Fibonacci Numbers. Edited by Marjorie Bicknell and Verner E.
Hoggatt, Jr. FA, 1972
Fibonacci's Problem Book, Edited by Marjorie Bicknell and Vemer E. Hoggatt,
Jr. FA, 1974.
The Theory of Simply Periodic Numerical Functions by Edouard Lucas. Translated from the French by Sidney Kravitz. Edited by Douglas Lind. FA, 1969.
Linear Recursion and Fibonacci Sequences by Brother Alfred Brousseau. FA,
1971.
Fibonacci and Related Number Theoretic Tables. Edited by Brother Alfred
Brousseau. FA, 1972.
Number Theory Tables, Edited by Brother Alfred Brousseau. FA, 1973.
Recurring Sequences by Dov Jarden. Third and enlarged edition. Riveon Lematematika, Israel, 1973.
Tables of Fibonacci Entry Points, Part One. Edited and annotated by Brother
Alfred Brousseau. FA, 1965.
Tables of Fibonacci Entry Points, Part Two. Edited and annotated by Brother
Alfred Brousseau. FA, 1965.
A Collection of Manuscripts Related to the Fibonacci Sequence ~- 18th Anniversary Volume. Edited by Verner E. Hoggatt, Jr. and Marjorie BicknellJohnson. FA, 1980.
Please write to the Fibonacci Association, University of Santa Clara, CA 95053,
U.S.A., for current prices.